


HANDBOOK OF OPT 



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LIBRARY OF CONGRESS. 

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Chap. Copyright No. 

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UNITED STATES OF AMERICA. 



HANDBOOK OF OPTICS 




•Jj^7^^° 



HANDBOOK OF OPTICS 



FOR 



STUDENTS OF OPHTHALMOLOGY 



WILLIAM NORWOOD SUTEE, B.A,, M.D. 

PROB'ESSOR OF OPHTHALMOLOGY, NATIONAL UNIVERSITY, AND 

ASSISTANT SURGEON, EPISCOPAL EYE, EAR, AND 

THROAT HOSPITAL, WASHINGTON, D.C. 



THE MACMILLAN COMPANY 

LONDON: MACMILLAN & CO., Ltd. 

1899 

All rights reserved 



3948a< 



COPYRIGHT, 1899, 

By the MACMILLAN COMPANY. 






.101 a., 1- • 




Korhjooti ^vegg 

J. S. Gushing & Co. — Berwick & Smith 
Norwood Mass. U.S.A. 






PREFACE 

In the following work there is presented so much 
of the science of optics as pertains directly to oph- 
thalmology. Simplicity has been sought so far as 
this is not incompatible with thoroughness ; for 
whoever would become versed in ophthalmology as 
a science must in the beginning make the mental 
effort necessary to acquire a clear understanding 
of the refraction of light through a compound opti- 
cal system such as the eye. 

The demonstrations, some of which may appear 
formidable to the student, require no knowledge 
of mathematics beyond that of simple algebraic 
equations and the elementary truths of geometry. 
For those who may not be familiar with the trigo- 
nometrical ratios, a brief synopsis has been fur- 
nished in an appendix. 

In demonstrating refraction by prisms and by 
spherical surfaces. Heath's " Geometrical Optics " 
has been used as a basis, but many modifications 
have been made. 



vi PREFACE 

A uniform notation, with which the student will 
easily become familiar, has been preserved through- 
out the book ; by this means those who may be 
indisposed to follow the algebraic processes in 
detail will be aided in understanding the methods 
of demonstration. 

Washington, June, 1899. 



CONTENTS 



CHAPTER PAGE 

1 



Introduction 

I. Refraction at Plane Surfaces 
11. Refraction at Spherical Surfaces 

III. Refraction through Lenses . 

IV. The Eye as an Optical System 



6 
26 
40 
62 



Y. The Determination of the Cardinal Points 

OF THE Eye in Combination with a Lens 90 

YI. Errors of Refraction — Lenses used as 

Spectacles 94 

YII. The Effect of Spherical Lenses upon the 

Size of Retinal Images .... 109 

YIII. Cylindrical Lenses 124 

IX. The Twisting Property of Cylindrical 

Lenses . . . . . . . . 138 

X. The Sphero-Cylindrical Equivalence of Bi- 

CYLINDRICAL LeNSES . . . . . 148 

XI. Oblique Refraction through Lenses . . 163 

XII. The Effect of Prismatic Glasses upon 

Retinal Images . . . . . . 169 

vii 



Viii CONTENTS 

CHAPTER PAGE 

XIII. The Reflexion of Light 177 

XIV. The Optical Principles of Ophthalmometry 

AND OF Ophthalmoscopy . . . . 184 

Appendix I . . 198 

Appendix II . . 200 

Index . . 205 



HANDBOOK OF OPTICS 



INTRODUCTION 

That branch of physical science which treats of 
light and vision is called Optics. It may be sub- 
divided into Geometrical, Physical, and Physiological 
optics. Geometrical optics deals with the theory of 
light ; it is a '' mathematical development " of the 
experimental laws by which light is supposed to be 
controlled, — the laws of reflexion and refraction, 
and the supposition that light travels through homo- 
geneous media in straight lines. Physical optics 
investigates the causes and nature of light ; while 
Physiological optics treats of the phenomena of vision 
or the sensation produced by the action of light 
falling upon the retina. 

Catoptrics and Dioptrics, terms less used now than 
formerly, refer respectively to the phenomena of 
reflexion and refraction of light. The science of 
optics was practically unknown to the ancients. 
They had observed that light travels in a homogene- 
ous medium in straight lines, and they also knew the 
simple law of reflexion and the focusing property of 

1 



2 HANDBOOK OF OPTICS 

lenses and mirrors ; but their ideas of vision were 
most crude, it being commonly supposed that light 
was something given out from the eye. Strange to 
say, this theory did not entirely disappear for many 
centuries. 

Spectacles of spherical lenses were probably intro- 
duced in the thirteenth century. To a spectacle 
maker, Hans Lippershey, is ascribed the first tele- 
scope in 1608, and in the following year Galileo 
independently constructed his telescope ; but with 
the astronomer Kepler, who died in 1630, begins the 
true science of optics. Willebrod Snellius, professor 
of mathematics in Leyden, discovered the law of 
refraction ; and to Sir Isaac Newton is due the dis- 
covery that white light is composed of various 
colors, capable of separation by the action of a 
prism. 

Up to this time the propagation of light was 
thought to be instantaneous. Romer, a Danish 
astronomer, discovered in 1676 that time is re- 
quired for the transmission of light. This he 
inferred from discrepancies between the calculated 
and actual time in the observation of eclipses of 
Jupiter's satellites ; and he rightlj^ attributed these 
discrepancies to the unequal distances through 
which light had to travel, owing to the varying 
distance between the planet and the earth. Ter- 
restrial measurement of the rate of propagation of 



INTKODUCTION 3 

light was not accomplished until the middle of the 
present century. This was done with instruments 
of great precision by Fizeau and Foucault, and more 
recently by Professor Newcomb. The velocity as 
thus determined is 300,000,000 metres or 186,000 
miles per second. 

The question as to the method of transmission of 
light has been the subject of much controversy. 
Leaving aside the speculations of the ancients, the 
two theories are the Corpuscular or Emission The- 
ory and the Wave Theory. The first supposes a 
luminous body to give off certain particles, which, 
striking the eye, produce vision. The wave theory 
supposes that all space is pervaded by a substance 
called ether ; and that by means of this substance, 
waves, excited in the luminous body, are transmitted 
to the eye. While some of the phenomena of light 
can be explained by either theory, experiments by 
Huyghens, Young, and Fresnel have rendered the 
emission theory untenable. While we are forced to 
the belief that light is propagated in waves, we are 
ignorant as to the nature of these waves. It was 
formerly supposed that the ether was a highly elastic 
body, transmitting vibrations of its particles through 
space just as a rod of steel, if struck near one end, 
will convey the vibrations to the other end. 

To explain the phenomena of light it is necessary 
to suppose the vibrations transverse to the direction 



4 HANDBOOK OE OPTICS 

of propagation, as is the case in the illustration 
cited. The waves of sound travelling through air 
differ from these in that in the case of the sound 
waves the particles of air vibrate to and fro in the 
direction of propagation. 

The modern study of electricity has changed our 
conception of ether waves ; for according to the 
electro-magnetic theory of waves, each particle of 
ether is ''polarized" or charged with energy, which 
in turn is transmitted to the next particle, and so on. 
These waves of energy produce various effects de- 
pending upon the rapidity of vibration. Those of 
least rapidity are manifested as electricity, and next 
in order come heat-producing waves. As the rapid- 
ity increases we have light of different colors, red 
being the color of least rapidity of vibration. Be- 
low this color in the spectrum are found heat 
waves. Violet is the color of greatest rapidity, 
and beyond this no light will be seen, but certain 
chemical effects produced there indicate the presence 
of waves capable of causing chemical action. It is 
also thought probable that gravitation is exercised 
by the transmission of energy through ether waves. 
For the further study of the wave theory, which is 
necessary for the proper understanding of interfer- 
ence phenomena and of polarized light, the student 
is referred to complete treatises on the theory of 
light. 



INTRODUCTION 



The investigations in the following chapters are 
based upon the experimental laws of reflexion and 
refraction, and upon the supposition that light 
travels in a homogeneous medium in straight lines. 



CHAPTER I 

REFRACTION AT PLANE SURFACES 

A luminous body emits light in all directions. 
That portion of light which travels along a partic- 
ular line is called a ray. A collection of rays which 
do not deviate far from a central fixed ray is called 
a pencil. When a ray of light passing through a 
median! meets another medium of different den- 
sity, it is divided into two portions ; a part of 
the light is reflected back into the first medium, 
while the remaining portion passes into the second 
medium, and is generally altered in direction. 
Of the reflected light, a part is said to be regu- 
larly reflected and a part scattered. Strictly speak- 
ing, all reflexion is regular ; but owing to the un- 
evenness of the surface, the light is reflected in 
various directions. It is by this means that we see 
a non-luminous object ; for if the light were all 
regularly reflected, we should see only the image 
of the illuminating source. The more even the sur- 
face, the greater is the regularly reflected and the 
less the scattered light. If the substance is opaque, 
no light passes into it ; the incident light is either 

6 



REFRACTION AT PLANE SURFACES 7 

reflected or absorbed. When a ray passes from one 
medium to another, the two portions of the ray 
before and after entering the new medium are 
called, respectively, the incident and the refracted 
ray ; and the acute angles which they make with the 
normal to the surface are called, respectively, the 
angle of incidence and the angle of refraction. In 




Fig. 1. 



Fig. 1 BO is the incident ray, OR^ is the refracted 
ray, GOR is the angle of incidence, and DOR' is 
the angle of refraction. It is found by experiment 
that the incident and refracted rays always lie on 
opposite sides of the normal to the refracting sur- 
face ; that the angles of incidence and refraction 
always lie in the same plane ; and that the sine of 
the angle of incidence always bears a fixed ratio to 
the sine of the angle of refraction. This ratio, while 
fixed for the same two media, varies with the nature 
of the refracting substances. It is called the refrac- 



8 



HANDBOOK OF OPTICS 



tive index for the two media. This law of refrac- 
tion is called SnelPs law ; it is sometimes called 
Descartes' law, since he first published it in its 
present form. For many years prior to Snell's dis- 
covery, investigators had constructed tables express- 
ing the relation between angles of incidence and 
refraction, but even Kepler was unable to deduce 
from these the law governing this relation. This 




Fig. 2. 



law, discovered by experiment, acquires new interest 
from its corroboration of the wave theory of light. 
It was supposed — and modern experiments have 
proved it to be true — that light travels with dif- 
ferent velocities in media of different densities. Let 



REFRACTION AT PLANE SURFACES 9 

AB^ Fig. 2, be a small portion of the front of a 
wave of light which proceeds from a distant point ; 
then AB^ being an indefinitely small arc of a circle, 
is indistinguishable from a straight line, and may 
be regarded as such. AA^ represents the surface 
of refraction ; v the velocity of light in the first 
medium ; v' the velocity in the second medium. 
Then if t is the time required by the light to trav- 
erse the distance BA^^ we have BA^ = v - t^ Avhich is 
also equal to AI). If the wave had been unob- 
structed by the second medium, it would occupy the 
position A'ND at the end of the time t. But the 
portion of the wave front at A^ meeting the more 
dense refracting substance, does not travel so fast as 
the portion BA^ , The point A becomes the centre 
of a wave disturbance, which in the time t has 
reached the point C ; and, consequentl}^ AC = v' • t. 
Similarly, the portion of the wave at P travels in the 
second medium the distance P3I^, while it would 
have travelled in the first medium the distance PiV. 
Therefore, AO:AI) = v^:v, and FM^ : PN= v^iv, 
from which 

AC v^ PlSr AC AD AA^ 

or, 



AD V PN PM^ PN PA' 

From this equation it follows that PA' M' and 
AA' C are similar triangles, and consequently M 
lies on the line A'C. Since in like manner any 



10 HANDBOOK OF OPTICS 

other point of the wave front will at the end of the 
time t lie on the line A' C^ which is perpendicular to 
AC, then A^ O will represent the wave front after 
refraction, and AO will represent the direction in 
which the refracted light travels. 

If z be the angle of incidence of the ray, then 
BA'A = 90 - z, and AA'D = i. If r be the angle of 
refraction, then A' AC = 90 — r, and AA' C = r. 

From the triangle A' AD we have 









sin i = 


AD 

AA!' 


and from AA^ C 


we have 










sin r = 


AC 
' AAl' 


from 


which 


sin i 
sin r 


AD 

AC 


V 

v' 



From this we see that the constant ratio between 
the sines of the angles of incidence and refraction is 
that of the velocity of light in the first medium to 
the velocity in the second medium. We learn from 
experiment that the deviation is toward the normal 
to the surface when the ray passes from a rarer to a 
denser medium, and away from the normal when the 
ray passes from a denser to a rarer medium. If the 
deviation is toward the normal, then i is greater 
than r, and consequently v is greater than v\ Thus, 



REFRACTION AT PLANE SURFACES 



11 



according to the wave theory, the velocity of light 
must be greater in air than in a dense mediun:i, 
such as water or glass. On the other hand, accord- 
ing to the emission theorj^, the velocity must be 
greater in a dense medium than in a rare one.* Bat 
experiments have proved that the velocity is less in 
dense than in rare media ; and the evidence is in 
favor of the wave theory. 




Fig. 3. 



When light passes from a rarer to a denser me- 
dium, n is greater than unity, and since sin r = , 

sin r is never greater than unity whatever may be the 
angle i ; but when light passes from a denser to a 
rarer medium, n is less than unity, and, for certain 
values of i, sin r may become greater than unity. As 

* Preston's "Theoiy of Light," 2d ed., p. 17. 



12 HANDBOOK OF OPTICS 

the sine of an angle cannot be greater than unity, 
this would be an impossible value for r. It is found 
by experiment that when i has such value as to make 
sinr greater than unity, light does not pass out of 
the denser medium, but is reflected back into this 
medium. This is called the total internal reflexion, 
and the angle of incidence which makes sin r equal 
to unity is called the critical angle. A glance at 
Fig. 3 will show the meaning of this. It is also 
found, as we might expect from this phenomenon, 
that as the angle of incidence increases the propor- 
tion of reflected light increases, while that of re- 
fracted light diminishes. Advantage is taken of 
the phenomenon of total internal reflexion in the 
construction of certain optical instruments. 

Another experimental fact is the reversibility of 
the path of light, that is, if the direction of a ray be 
reversed so that the angle of refraction becomes the 
angle of incidence, then the original angle of inci- 
dence will become the new angle of refraction. This 
being so, it is evident that a ray after refraction 
through a medium with parallel surfaces will, upon 
reentering the original medium, be parallel to it^ 
direction before refraction. It will, however, un- 
dergo a lateral displacement varying with the thick- 
ness of the medium. See Fig. 4. 

We have learned that = n, where n is the 

smr 



KEFRACTION AT PLANE SURFACES 



13 



ratio of the velocity of light in the first medium to 
its velocity in the second medium. If the velocity 
in a vacuum is taken as the standard, the ratio of 
the velocity in any medium to the velocity in a 
vacuum is called the absolute refractive index. The 
ratio of the velocities for any two media may there- 
fore be expressed in terms of the absolute indices of 
the two media. Thus, if n be the absolute index 




Fig. 4 



of the first medium and n' that of the second, the 



n 



relative index for the two media will be — . Snell's 

n 

law thus becomes n • sin i = 7i' • sin r, and it is usually 
written in this form. 



14 HANDBOOK OE OPTICS 

In any refraction the greater the angle of inci- 
dence, the greater will be the deviation ; and the 
greater the angle of incidence, the greater will be 
the increase in deviation for a fixed increase in the 
angle of incidence. This follows directly from the 

equation - — = n. Reference to Appendix I. will 
sm r 

render this clear, for it is there shown that the sine 
of an angle increases less rapidly than the angle ; 
and the greater the angle, the less will be the change 
effected in its sine by a fixed increase of the angle. 
Hence, if r be smaller than z, a smaller increase in 
r will be required to maintain the constant ratio 
between sin i and sin r than in the greater angle i; 
and this is true in a greater degree as i approaches 
90 degrees. Since the deviation is expressed by 
i — r, it follows that this increases when i increases. 
As the path of light is reversible, the same holds 
true when i is less than r, that is, when the ray 
passes from a denser to a rarer medium. 

A medium bounded by two plane faces meeting in 
an edge is called a prism. At present we shall con- 
sider only the refraction of rays through prisms, 
reserving for a future chapter the more difficult sub- 
ject of refraction of peficils of light. We shall sup- 
pose the rays to lie in a principal plane of the prism, 
that is, in a plane which is perpendicular to the edge 
of the prism, and consequently to the plane of eacli 



REFRACTION AT PLANE SURFACES 



15 



face of the prism. We shall also confine our atten- 
tion to prisms whose refractive index is greater than 
that of air. When a ray of light passes through 
such a prism, the deviation is in all cases from the 
apex toward the thicker part of the prism. AVe 
have seen that a ray, passing through a plane 




Fig. 5. 



bounded by parallel surfaces, emerges without devia- 
tion. Let PQRS (Fig. 5) represent a ray passing 
through the medium ABOD^ AB and CD being 
parallel. RN is normal to the face CD, Now sup- 
pose the face CD be turned into the position ^D^ 
making the prism BAD^; then the normal B]V must 
turn into the position BW; and by this change the 
angle of incidence is increased from NRQ to N^ BQ. 



16 



HANDBOOK OF OPTICS 



We have learned that with an increase of the angle 
of incidence there is also an increase of deviation. 
KRS represents the deviation at the second surface 
when the face has the position (7i>, and as tlie devia- 
tion is increased by turning the face into the position 
AD' ^ then KRS' ^ greater than KRS^ will represent 
the deviation at the second face of the prism BAD' , 
RS is parallel to PQ, the direction of the ray be- 




FiG. 6. 



fore entering the prism ; hence SRS' represents 
the deviation of the ray in its passage through the 
prism, and this deviation is away from the apex of 
the prism. 

Light from an object at P (Fig. 6) would be so 
deviated as to enter an observer's eye at E, and the 
object would appear to be at P^ Hence an object 
seen through a prism is displaced toward the apex 
of the prism. 

All light is not equally deviated by prisms. If a 



REFRACTION AT PLANE SURFACES 17 

narrow beam of simliglit be passed through a prism 
in a darkened room, and the refracted light be inter- 
cepted by a screen, it will be found that the beam 
has been decomposed into bands of colored light. 
These are violet, indigo, blue, green, yellow, orange, 
and red. Of these, violet is most and red least 
deviated. They are called the colors of the spec- 
trum. Since the deviation of a ray by refraction is 
explained by the theory that the passage of light is 
retarded upon entrance into a substance of greater 
density, it is necessary to suppose that this retard- 
ing power is different for different colors ; violet, 
which is most deviated by the prism, must suffer the 
greatest retardation, and red, the color of least 
deviation, must be least retarded by the prism. It 
is supposed that only dense substances offer this un- 
equal resistance to the passage of light of different 
colors, and that in space and in air all light travels 
with the same velocity.* 

The property which prisms possess of separating 
colors is called dispersion. It is a most important 
property, but only incidentally concerns the student 
of ophthalmology. The chief use of prisms in oph- 
thalmological practice is for the purpose of changing 
the apparent position of objects. In the weaker 
prisms dispersion is not noticeable ; but if an opaque 

* Preston's ^' Theory of Light," 2d ed., p. 97. 



18 



HANDBOOK OF OPTICS 



object be viewed through a prism of considerable 
deviating power, it will be tinged with red toward 
tlie apex and with violet toward the base of the 
prism. 

Let PQBS (Fig. 7) represent a ray passing 
through the prism whose apex is at (9, and whose 
faces are inclined at an angle a. Tins angle is 
called the refracting angle of the prism. At Q and 




Fig. 7. 



R draw the normals i^7!f and LRN. Let the angle 
of incidence PQM be called ^, and the angle of 
refraction LQR be called r. Also let LRQ be r\ 
and NRS be e ; then from the law of refraction we 
have sin i = n - sin r, and sin e = n * sin r'. The 
angle ORQ is equal to 90 — r' and OQR is equal to 
90 — r. Since the sum of the three angles of the 
triangle ROQ must be equal to 180 degrees, we 
have 

a + 90 - r + 90 - / = 180 ; or, r + / = a. 

The deviation of the ray at the first surface is 



EEFK ACTION AT PLANE SURFACES 19 

represented by i — r, and at the second surface by 
e — T^ , The total deviation is denoted by z + e — 
(r + /), OY i -{- e — a. 

Hence the deviation produced by a prism is equal 
to the sum of the angles of incidence and emer- 
gence, minus the refracting angle of the prism. 

Let us suppose that the ray passes symmetrically 
through the prism, that is, that the angles of inci- 
dence and emergence are equal. The angle of 
incidence ^ is greater than the angle of refraction r, 
since the index of the prism is greater than that 
of air. Hence, as was shown on page 14, when i 
increases, r also increases, but less rapidly than i. 
Since in the triangle LRQ the angle L remains con- 
stant, then when r increases, r^ must undergo a cor- 
responding decrease, for the sum of the three angles 
of the triangle must be 180 degrees. From the 
equation sine = ti • sin/, it follows that if r^ de- 
crease, e must decrease more rapidly than r^ There- 
fore, if we start with the ray which passes sym- 
metrically through the prism, and increase the angle 
of incidence, the effect will be to increase the devia- 
tion at the first face of the prism and to diminish it 
at the second face ; but as r is now greater than r^ 
the increase at the first face outbalances the de- 
crease at the second face, and the total deviation is 
increased. If we trace this ray backward, we see 
the effect of making the angle of incidence smaller 



20 HANDBOOK OF OPTICS 

than that of the symmetrical ray, that is, in this case 
also the deviation is increased. Hence the symmet- 
rical ray is the ray which undergoes the least devia- 
tion ; it is called the ray of minimum deviation. 

If D denote the deviation of a ray in passing 
through a prism, I) =• i + e — a, from which e = a + 
D-i, 

We have seen also that a = r + r^ from which 
r' = a — r. 

Substituting these values of e and r' in the equa- 
tion 

sin e 

- — j = n, 

, sin (a + D — i) 

we nave > — = n^ 

sni (a — r) 

or sin [(a + D) — ^] = 7^ • sin (a — r), or sin (a + i)) 
cos i — cos(a + J)) sin i = n (sin a cos r — cos a sin r). 
When the angle of the prism is small, sin (a + i>) 
and sin a do not differ materially from the angles 
a + D and a.* Likewise, it is easily seen that the 
cosines of these small angles do not differ materially 
from unity. Making these substitutions, we have 

* The measurement of an angle is expressed by the subtending 
arc divided by the radius of this arc ; it is readily seen that this is 
practically equivalent to the sine of the angle when the angle is very 
small. 



REFRACTION AT PLANE SURFACES 21 

(a + D) cos i — sin i = n - a cos r — n - sin r; 
or, 

J) . cos i= a(n ' cos r — cos 0, since sin i = n • sin r. 



Hence D = a{ -, 1 

cosz 



When the ray passes nearly perpendicularly through 
the prism, as does the ray of minimum deviation 
in a prism of slight deflecting power, then cos r and 
cosz are both very nearly equal to unity. In this 
case the deviation is approximately equal to a(ii — V), 
If the index of refraction of the material of which 
the prism is made is 1.5, as is approximately true 
of spectacle glass, the deviation becomes equal to -. 

The exact index for glass is greater than 1.5; its 
average index may be regarded as 1.53, and there- 
fore the deviation even in weak prisms is more than 
one-half the refracting angle of the prism, but for 
practical purposes the two may be considered equal. 
In prisms of high deviating powder the deviation 
is perceptibly greater than one-half the refracting 
angle. 

Prisms in the oculists' trial case are usually num- 
bered in degrees of the refracting angle, or accord- 
ing to the deviating power in the position of minimum 
deviation, the latter method having been first sug- 



22 HANDBOOK OF OPTICS 

gested by Dr. Edward Jackson,* of Philadelphia, 
as being more scientific than the old notation in 
degrees of the refracting angle. Other systems of 
numbering prisms have also been advocated and are 
to some extent used. The units in these systems 
are : the centrad, introduced by Dennett ; f the 
prism-dioptre, introduced by Prentice ; J and the 
metre-angle, introduced as a measure of convergence 
by Nagel, and as a prism unit by Maddox.§ 

It is sometimes desirable to know the result of 
combining two prisms whose edges are not parallel. 
We know that prisms deviate light in a direction 
at right angles to the edge of the prism. Hence 
when two prisms are combined so that their edges 
are not parallel, the deviating power of the second 
prism must be added to that of the first; but the 
direction in which the power is exerted is not the 
same in the two prisms. 

In Fig. 8, let AB and AD represent the direc- 
tions in which light is deviated by the first and 
second prisms respectively ; also let the length AB 
represent the displacement of a ray of light which 
the first prism produces, as measured on a screen 
at a fixed distance from the prism, and let AD 

* Trans. American Ophth. Soc, 1887 and 1888, 

t Ibid, 1889. 

} Archives of Ophthalmology, 1890. 

§ Wood's "Med. and Surg. Monographs," Vol. IX., No. 2. 



REFRACTION AT PLAXE SURFACES 23 

represent the displacement wliicli the second prism 
produces at the same distance. Then to find the 
combined effect of the two prisms, it is only neces- 
sary to construct the parallelogram ABCD^ and 
the diagonal AC will represent the direction and 
the length of displacement produced by the two 
prisms acting together ; for the problem is the same 




.^^ / 



..^ 



/ 

I 

/ 

/ 

/ 



/ 

/ 



/ 



Fig. 8. 

as that in which an object at A is acted upon by a 
force which would move it from A to B^ and at the 
same time by a force which would move it from A 
to i), the result being that the object is moved from 
A to (7. 

It would, however, be inconvenient to find the 
linear displacements AB and AD ; and to avoid this 
we must use a relation which exists between the 
angular deviation and the linear displacement. 

Let d represent the angle of deviation of the first 
prism, and dJ that of the second prism ; then, since 



24 HANDBOOK OF OPTICS 

an angle is measured by its subtending arc divided 
by the radius of this arc, we have 



■J arc -, J, arc' 
a = — and a' = — -, 



If the distance of the screen from the prism be the 
radius, then the arc described with this radius and 
terminating in A and B will be the subtending arc 
of the angle d. When d is small, the aic and the 
straight line AB will be so nearly equal that we may 
consider them identical. Similarly the arc of the 
angle 6?^ when this angle is small, may be replaced 
by the line AI). Thus for small angles we may with 
any unit lay off AB so that it contains as many units 
as the arc of the angle d contains degrees, and with 
the same unit lay off A Z) so that it contains as many 
units as the arc of the angle d^ contains degrees ; 
then AC SiS measured by the same unit will give the 
number of degrees of deviation produced by the 
equivalent prism, and AO will represent the direc- 
tion of deviation of this prism. The angle BAB is 
the angle of inclination of the two prisms, for BA 
and AD are perpendicular to the edges of the first 
and second prisms respectively. The angle ABC is 
equal to 180 — BAB. Hence in the triangle ABC 
we know the two sides AB and BC and the included 
angle ABC. We can find the third side ^6^ and the 



REFRACTION AT PLANE SURFACES 25 

angle OAB, which the equivalent prism must make 
with the first prism. 

The side J. (7 is obtained from the equation 

(^(7)2 = QABy + QBOy - 2 AB X BO cos ABC. 

Having found A (7, the angle CAB is found from the 
equation 

sin CAB ^BC 

sin ABC AC' 

since in any triangle the sines of the angles are pro- 
portional to the opposite sides. If the tw^o prisms 
are at right angles, then 

(ACy = (ABy + (BCy, and tan CAB = ^. 



CHAPTER II 

REFRACTION AT SPHERICAL SURFACES 

Let (Fig. 9) be the centre of a spherical surface, 
separating two media whose refractive indices are 
respectively n and n' . 

A ray of light, PB, meets the surface at jB, and is 
refracted so as to assume the direction RQ, NRP 
is the angle of incidence, and ORQ is the angle of 
refraction. Then we have n • sin NRP = n' - sin ORQ, 

From the triangles ORP and ORQ we have 

sin (180 -iVTgP) ^ sin NRP ^ OP .^. 

^'mROA siiiROA RP ^ ^ 



sin ORQ ^ sin ORQ ^ OQ 
sin (im-ROA) sin ROA RQ 

Dividing (1) by (2), we have 



(2) 



sin NRP ^qP^ RQ_ ^3^ 

sin ORQ RP OQ ^ ^ 



The first member of (3) is equal to — 

Therefore «-|5=^'-||- 

26 



KEERACTION AT SPHEMCAL SURFACES 27 

The ray PA is normal to the spherical surface at 
A ; hence it will undergo no deviation, but will con- 
tinue in the same straight line, PAQ. 

This line is called the axis of the refracting sur- 
face. The ray which meets the surface at R meets 
the axis at Q. If every other ray of the pencil which 
is refracted at the spherical surface should meet the 
axis at Q, this would be the focus of the pencil after 
refraction. A study of Fig. 9, however, will show 




that this will not be so. If the arc AR be revolved 
around the axis, a portion of a sphere will be gener- 
ated, and all rays meeting this surface at the distance 
AR from the axis will meet at Q. If we take the rays 
which meet the spherical surface at a distance ATt\ 
they will not meet the axis at Q» The relations 
between ROA and the angles of incidence and re- 
fraction are such that as ROA increases, OQ must 
diminish ; and consequently those rays which meet 
the refracting surface more remotely from A inter- 



28 HANDBOOK OF OPTICS 

sect the axis at a less distance from A than do those 
rays v>^hich lie near the axis.* 

The rays BQ and M^ Q' meet at q. The curve 
formed by the locus of all such intersecting points is 
called the caustic of the refracting surface. An illus- 
tration of a caustic can be seen by placing a glass of 
water on a table so that sunlight strikes the glass. 
The brightly illuminated curve which will be noticed 
represents the intersection of rays after refraction by 
the water. The property of spherical refracting sur- 
faces, by which rays of a pencil proceeding from a 
point meet the axis at different points after refrac- 
tion, is called spherical aberration. The nearer that 

— — - approaches a constant quantity, the less is the 

spherical aberration. It is easily seen that this condi- 
tion is best fulfilled when H is near the axis, and when 
the curvature of the surface is slight as compared with 
the distances PA and AQ, The greater the lengths 
PA and AQ^ the farther may It be removed from the 
axis withou.t appreciable spherical aberration. Spec- 
tacle lenses, being lenses of long focal length, do not 
cause much aberration. In the eye the focal distances 
are short ; but, owing to the cutting off by the iris 
of all peripheral rays, and to the peculiar construction 
of the crystalline lens, aberration is not appreciable. 
Since the more peripheral rays are too strongly 

* Heath's " Geometrical Optics," 2d ed., p. 144. 



KEFRACTION AT SPHERICAL SURFACES 29 

refracted in comparison v/ith those near the axis, it 
is evident that the curvature of the spherical surface 
is too great for peripheral rays. A surface with 
diminishing curvature, such as an ellipsoid, Avould 
consequently produce less aberration than the spheri- 
cal surface. The curve which produces no aberra- 
tion is known to mathematicians as the Cartesian 
oval. Such surfaces, however, are not in practical 
use ; for, by suitable combinations of lenses, aber- 
ration can be almost entirely overcome, even in the 
lenses of very short focal distance and wide aperture 
used in microscopes. 

Besides spherical aberration, we have also chro- 
matic aberration, v/hich is due to the unequal deviating 
power of the refracting surface for different colors. 
In the construction of microscopes and other delicate 
optical instruments, the annulment of this defect is 
a matter of the utmost importance, but owing to the 
long focal length of spectacle lenses and to the pre- 
vention by the iris of all peripheral rays from enter- 
ing the eye, chromatic aberration does not attract 
attention in ordinary vision. 

All our formulae will be based upon the assump- 
tion that PA, HP, AQ. and RQ are great in com- 
parison with the curvature of the surface, and that 
onlj^ rays near the axis are allowed to pass into the 
refracting medium. With this understanding, our 
equation, 



30 HANDBOOK OF OPTICS 



becomes 



OP _ , OQ 
'"'PR-'' 'BQ' 



(FA + AO) ^ . (AQ-AO) 
FA AQ 



OF 

For, upon our assumption, — — will not differ 

materially from — — , and — ^ will not differ mate- 

riallv from — -^• 
AQ 

If PA be denoted by /, and AQ by/', and r be 

the radius of the spherical surface, the equation will 

become 

■ f f 

This equation may be reduced to the form, 

This is the relation between a point P and its 
focus Q. P and Q are called conjugate foci, and / 
and /' are conjugate focal distances. A pencil of 
light from P will be brought to a focus at Q^ and, 
conversely, a pencil proceeding from Q will be 
brought to a focus at P.* In our demonstration we 

* When Q is virtual, we must modify this clause so as to read : 
"A pencil directed towards Q will be focused at P." 



REFRACTION AT SPHERICAL SURFACES 31 

have considered all the quantities as positive. This 
is the convention of signs most convenient in dealing 
with lenses ; but in working out formulae for several 
refracting surfaces, it would lead to confusion. In 
these cases it is better to consider all quantities posi- 
tive when measured from left to right, and negative 
when measured from right to left. Thus, J-P, 
measured from A^ would be negative ; while AQ^ as 
in the former convention, would be positive. We 
should have to replace/ in our equation (a) by — /. 
Doing this, we have 

We shall use (ct) in demonstrating the properties 
of single lenses, but (5) will be more suitable in 
tracing the path of light through the several re- 
fracting media of the eye. 

n n^ _n' —n 

is the equation between conjugate foci for all posi- 
tions of / and /'. If, in this equation,, we make / 
infinite, we have 



n — n 



This means that if the rays proceed from a point at 
an infinite distance, that is, if the rays are parallel 



32 HANDBOOK OF OPTICS 

to the axis in the first medium, the value for /' is 
This is called the posterior or second princi- 



nr * 



?^^ — n 



pal focal distance, and the point where the rays meet 
the axis is called the posterior or second principal 
focus. Similarly, if f is infinite, that is, if rays pro- 
ceeding from a point, P, are parallel after refraction 

we have, as the corresponding value of /, 



9 
nr 



71^ — n 



The point P thus becomes the first or anterior prin- 

7)7* 

cipal focus, and the distance — is the first princi- 

n' —n 

pal focal distance. The anterior focal distance is 
denoted by the letter i^, and the posterior focal 
distance is denoted by _F^ 

If we divide equation (a) by , we have 

r 

nr . n^r 






(ji' — 7i)f (n' — 71) f 

This equation gives us the relation between two 
conjugate points and the two principal foci. 

F and / are positive when to the left of the refract- 
ing surface, and F^ and f are positive when to the 

* In the small pencils which enter the eye, the rays may be con- 
sidered parallel when they proceed from a point situated 6 metres 
from the eye ; hence / is intinite when it has not less than this 
length. 



EEFRACTION AT SPHERICAL SURFACES 33 

right of this surface. When nJ is greater than n, F 
and F' are both positive or both negative according 
as r is positive or negative ; and when 7i^ is less than 
71, F and F' are both positive or both negative ac- 
cording as r is negative or positive. In other words, 
F and F^ are both positive when the convex surface 
is turned toward the medium of less refractive in- 
dex, and both negative when the convex surface is 
turned toward the medium of greater index. 

We have seen that rays which are parallel to the 
axis in the first medium will, after refraction into 
a denser medium at a convex surface, meet in an 
actual focus in the second medium ; and we have 
also seen that F^ in this case is positive. On the 
other hand, if the refracting surface be concave, the 
rays will not meet in a point in the second medium ; 
they will be rendered divergent by the refraction, 
and their direction will be such that, if prolonged 
backward, they will meet in a point in the first 
medium. Such a focus is only imaginary^ not a real 
meeting point of the rays, as in the former case ; it 
is called a virtual focus. It is, as we have seen, nega- 
tive. Hence, with the convention of signs which we 
are using, real foci are positive, and virtual foci are 
negative. Figure 10 illustrates the virtual focus. 

A study of equation Qd) will show that if n^ be 
greater than n, and r be positive, then, so long as/ is 
greater than #, /' will be positive ; and if / be less 



34 HANDBOOK OF OPTICS 

than F^ /' will be negative. This indicates that if 
rays proceeding from a point on the axis are re- 
fracted into a denser medium at a convex surface, 
they will after refraction meet the axis in a real 
point so long as the point from which they proceed 
is farther from the surface than the first principal 
focus. If the point is within this focus they will 



\ 



1 



Fig. 10. 



not meet the axis after refraction, but if prolonged 
backward they will meet in a virtual focus, and this 
will be farther from the surface than the point from 
which the light proceeds. 

The following equations will also be found useful 
in our studies : 

Since F = ^^^^ and F' ^ ^'"^ 



n' — n n^ — n 



we have F : F^ =n : 'n! -, or. 



F^^n 

F' n'' 



Also, if P (Fig. 11) be the position of a point 
and Q its conjugate, F and F' the principal foci, then 



REFK ACTION AT SPHEUICAL SURFACES 



35 



FF = f-F^iid QF' = f' - F. Let PF be denoted 
by u and QF^ by u^ \ tlien/= u -\- F and/' = i^' + F\ 



Hence, 



F 



+ 17 



F^ 



u + F u' + F^ 
From which we deduce 

uu'=FF'. 



= 1 



Fig. 11. 



Let us next take a point P not on the axis AA^ . 
Draw POP' (Fig. 12) through the centre, 0, of the 
refracting surface. This line is the new axis and P' 
is the focus conjugate to P. If OP be equal to OA^ 




then OP' and OA' will be equal, and PA and P^A' 

will be arcs of circles. As RHR is revolved on the 
axis, producing the spherical surface, so the arcs PA 
and P'A' also produce portions of spherical surfaces. 



36 * HANDBOOK OF OPTICS 

If the radii OP and OP^ be great as compared with 
the length of these arcs, the portions of spherical 
surface which they generate will not differ materially 
from plane surfaces. 

Therefore the planes at A and A^ perpendicular to 
the axis AA^ are called conjugate focal planes. The 
image of an object lying in the plane at A will lie in 
the plane at J.^ A plane tangent to the refracting 
surface at H is called the principal plane of the 
refracting surface. If the lines AR and PW were 
terminated in such a plane, and from their points of 
intersection with the plane lines were drawn to A^ 
and P\ respectively, the resulting lines would very 
nearly coincide with those as drawn- in Fig. 12 ; 
and, for practical purposes, the two results might be 
considered identical. 

The point jET, where the principal plane and axis 
intersect, is called the principal point. 

We shall now show how we may construct the 
image of an object if we know the position of certain 
points. Let PA (Fig. 13) be the linear dimension 
of an object. If we can determine the point of inter- 
section after refraction of two rays from P, this point 
will evidently be the image of P. First we take a 
ray PFR , which passes through the anterior focus 
F \ then we know that this ray must after refraction 
be parallel to the axis AA\ It is represented by 
R^ P' . Next we take a ray Pi?, which before refrac- 



REFRACTION AT SPHERICAL SURFACES 



3T 



tion is parallel to tlie axis ; then after refraction it 
mnst pass through the second principal focns F' , It 
is represented by RP' , Then P\ the point of inter- 
section of tlie two rays from P, is the image of P. 
The image of all points in the line AP must lie in 
the line A' P' \ hence, A^ P^ is the image of AP. 

We can now appreciate the importance of these 
points and planes. They are called the cardinal 



> 


R 


^ 


^ 


. F^ 




^^^— ___p 


H 


A' 


' ^~^- 


^^ 


— ^l„, 




\ 






r 



Fig. 13. 



points and planes of the refracting surface. The 
centre of the refracting surface also possesses a dis- 
tinctive property, in that rays which pass through 
it undergo no deviation, since such rays are normal 
to the refracting surface. It is called the nodal 
point. The cardinal points of a single refracting 
surface are four ; namely, the ^:)r^?2c/paZ pointy the 
two principal foci, and the 7iodaI point. The cardinal 
planes are the principal plane and the two principal 
focal planes. 

If be the linear dimension of the object, and i 



38 



HANDBOOK OF OPTICS 



that of the image, then from the similar triangles 
PAF and HFB^ (Fig. 13) we shall have 

PA __ AF _u _ u ^ 

llBJ~"FH' """' ^"^' ""^^ ~i~~F' 

Also from the triangles RHF^ and F^A^F\ 






or, - = 



F[ 



From either of these equations we can determine 
the size of the image. We may also determine the 
size of the image in terms of the divergence of the 

""" — ^ — ^_ M 

-S 




Fig. 14. 

rays before and after refraction. Let MP (Fig. 14) 
be the linear dimension of an object. A ray from 
P, meeting the refracting surface at R, would be 
refracted so as to assume the direction RS\ and if 
prolonged backward, it would meet the axis at Q^ 
which is conjugate to P. Likewise, iVis conjugate 
to M, and QN is the image of PM, The angle 

* Since the image, when real, as in the figure, lies on the oppo- 
site side of the axis to the object, we must consider it negative. 



REERACTION AT SPHERICAL SURFACES 39 

RPA^ which we call a, expresses the divergence of 
the pencil before refraction, and RQA or a^ is the 
divergence after refraction. 

Then tan a = — — , and tan a^ = -—— ; 
I^A QA 

£ , . -, tan a QA f 

irom which = i--. = ^. 

tan a^ FA j 

We have also the equation 

PO_^PM 
QO QN 

From the law of refraction we have, as on page 26, 

PO , QO 
RP RQ 

or, since RP may for rays near the axis be replaced 
by/andi2^by/, 

PO , QO PO n^f 

1 PM n^f T n . n^ .^^ 

hence -— — or -=:—-, and o • - = z • — . (1) 

From this equation and from ^ == we deduce 

f tan a-^ 

-n ' tan a = i *n' - tan a^. (2) 

This is known as Helmholtz' formula. 



CHAPTER III 

KEFRACTION THROUGH LENSES 

Haying investigated the refraction of light at one 
spherical surface, we are now prepared to studj^ re- 
fraction through lenses. A lens is defined as a por- 
tion of a refracting substance bounded by two curved 
surfaces centred on the same axis. If the radius of 
curvature of one surface is infinite, then the corre- 
sponding surface is plane, and the lens is bounded by 
one curved and one plane surface. Ordinarily, the 
surfaces of lenses are spherical ; but lenses have 
been constructed whose surfaces were ellipsoidal or 
paraboloid. The onl}^ lenses in practical use, how- 
ever, are those whose generating curve is a circle ; 
and we shall confine our attention entirely to lenses 
of this nature. 

The thickness of a lens is the distance between the 
bounding surfaces as measured along the axis. 

A lens bounded by two convex surfaces is called a 
double convex or bi-convex lens ; one bounded by two 
concave surfaces is called a bi-concave lens. 

A lens, of which one surface is convex and one 
concave, is called a convexo-concave lens, or a menis- 

40 



REFRACTION THROUGH LENSES 41 

cus. Lenses of this form are used as spectacles, and 
are known as periscopic lenses. 

The terms plano-convex and plano-concave need no 
explanation. 

In our demonstration we shall take as the typical 
case the double convex lens ; and we shall consider 
the refractive index of the material of which the lens 
is composed to be greater than that of the air by 
which it is surrounded. The index of the lens will 
be indicated by the letter n, and that of the air by 
unity. 

We shall first show that there are two points on 
the axis of the lens which are useful in the deter- 
mination of the positions of conjugate foci. These 
points are a pair of conjugate foci, such that any 
incident ray directed toward one of them will, after 
refraction, appeal* to come from the other, and in a 
direction parallel to that before refraction. These 
points are called the nodal points of the lens. 

To find the position of these points, we draw any 
radius of the first surface, as OQ (Fig. 15). Next 
we draw a radius, 0' Q' , of the second surface, so that 
OQ and 0' Q' are parallel. Connect the points Q 
and Q^ by the straight line QQ\ which meets tlie 
axis at 0, Then from the similar triangles OCQ 
and 0'0Q\ we have OC: O'G = OQ: O^Q =tm\ 

from which = — 



42 



HANDBOOK OF OPTICS 



00 
Therefore, — — is a constant quantity, irrespective 
(/ o 

of the position of Q, from which the first radius is 
drawn ; and consequently must be a fixed point. 
The ray of light, RR\ which passes through (7, is, 
after refraction, parallel to its direction before re- 




FiG. 15. 



fraction ; because, the radii OQ and 0' Q^ being par- 
allel, the planes perpendicular to the curved surfaces 
at Q and Q^ are parallel, and the lens will, for this 
ray, act as would a piece of plane glass. 

Let N be the point on the axis tow^ard which the 
ray is directed before refraction. The ray is so re- 



REFRACTION THROUGH LENSES 43 

fracted at the first surface as to pass through C ; 
then C is conjugate to N in the first refraction. 
After refraction at the second surface, the ray ap- 
pears to pass through iV'; then N' is conjugate to Q 
in the second refraction. Since N is the virtual in- 
tersecting point of the ray and axis before refraction, 
and N' is the virtual intersecting point of the ray 
and axis after refraction, then N and N' are conju- 
gate points with respect to both refractions. 

The point Q is called the optical centre of the lens. 
All rays which pass through it are, after refraction 
through the lens, parallel to their direction before 
refraction. 

To find the position of the points iV, N' ^ and C we 
have the following relations : 

00^ = r + r' - e ; 

where r and r^ are the radii of the surfaces, and e is 
the thickness of the lens. Also, 

00 ^ r . 
0^0 r'' 

and consequently, 

00 r 00 r 

or, 



OO+O'O r + r^' ,^^r'-e r + r'' 
00 = -^ir + r'-e)=r--^, 



44 HANDBOOK OF OPTICS 



AC= OA- OC=r-r+ ^'^ ^"^ 



r + r' r + r 



Similarly, A^ C- 



ey 



r + r 



But since in the first refraction iVand are con 
jugate foci, we have from formula (a), page 30, 



In n — 1 



AJSr AO r 
Substituting the value of AC, this becomes 



1 , n(r + ^') n — 1 

1 — ^^ ^ = ' 

AJV er r 



If rays proceeding from F^ are parallel to the axis 
after I'efraction at the first surface, and these parallel 
rays after refraction at the second surface meet at 
i^2^ then 

AF. = -^ = F., and A^F, = -^ = F^. 

71 — 1 '^ 71 — 1 

Making these substitutions, we have 

1 <F, + i^,) , 1 , 



+ 



• AN eF^ J^i 
or, AN=^ - ^^ ^V7^ X ^■ 



REFRACTION THROUGH LENSES 45 

If e be replaced by n • c, this equation will become 



AN= 



cF, 



^1 + ^2 



Since AN is negative for the convex lens, N lies 
to the right of A, and since in the lenses which are 
in practical use, the numerator F^ is less tlian the 
denominator '/v(i<\ + F^— e, the length AN is less 
than 6, and the nodal point N lies within the lens. 
Similarly, we find that the nodal point N' lies within 
the convex lens, and that its distance from A' is 

cF, 



F,+F,^-c 

Using the same equation, but changing the sign 
of r, we should find that in bi-concave lenses, also, 
tlie nodal points lie within the lens. 

To repeat, the nodal points are two conjugate 
points such that a ray of light directed toward one 
of them is so refracted as to pass through the optical 
centre of the lens, and emerges in a diivQciiow parallel 
to that before refraction, Sucli a ray undergoes only 
a lateral displacement due to the thickness of the 
lens. 

We shall now demonstrate the method of finding 
the focus Q, after refraction by a lens, of a pencil 
of light from a point P on the axis of the lens. 
The ray PR (Fig. 16) meets the first refracting 



46 



HANDBOOK OF OPTICS 



surface at jB ; it is refracted toward R' ^ and, if 
prolonged, it would meet the axis at Q\ But, after 
travelling the distance RR, it meets the second 




Fig. 16. 



surface of the lens and is refracted to Q. Then in 
the first refraction Q' is conjugate to P, and we 
have from equation (^i^), page 80, 

1 n _ ^ — 1 . 

PZ Z^~ r ' 

or, substituting / for PA^ f for AQ ^ and F^ for 

r 
n— 1 



l + ^ = -l 



(1) 



Also in the second refraction Q is conjugate to 
Q' \ therefore, 

n 1 _ ^ — 1 . 

or, substituting /^^ for A! Q\ f^ for A' Q, and F.^ for 



n—1 



, we have 



// /i ^2' 



(2) 



REFRACTION THROUGH LENSES 47 

Reference to the figure will indicate that AA^^ 
the thickness of the lens, is equal to the difference 
in length between'/^ and f-^' ; but we also notice 
that /' and f^' have opposite signs, that when f is 
positive, /j' is negative. Hence the difference be- 
tween /' and /-^^ will be expressed algebraically by 
/' +/i' ; and if e represent the thickness of the lens, 
/^ +/^' = ^, or 7ic^ n being the refractive index of 
the lens, and c being of such value that e is equal 
to nc. From these equations we can determine the 
relation between P and Q. From (1) we obtain 

f'= " 



JL_1 

F, f 
n 



From (2), // = -^ ^. 

Substituting these values in the equation /^+/^' 
= nc^ we have 

n n 

+ -q T = lie. 



F, f F, f. 

This, by reduction, becomes 

-F,JiF,-c)^cF,F,..: (4) 



48 HANDBOOK OF OPTICS 

This equation is true for all values of/ and /j. If 
we make / infinite, the corresponding value of/^ will 
give us the focus for rays parallel to the axis before 
refraction. 

Dividing equation (4) by / and making / iufinite, 
we derive 

Similarly, if /^ be infinite, that is, if the rays be 
parallel to the axis after refraction, we shall obtain 



^ F^ + F,^- 



c 



The points determined by these equations are the 
prmcipal focal points of the lens. They are not 
usually measured from the surfaces of the lens, how- 
ever, but from the points iV^and iV^ It will be seen 
that as thus measured the two focal distances are 
equal. 

f I A,Y,^F 2(F^-o-) . cF, _ F,F, 

-^^^ Fi + F,^-c^F^ + F,^-c Fi + F,^-c' 

Hence in a lens tlie two principal focal distances are 
equal, and this distance is found from the equation 

F= ^^ , 

F. + F^-c' 



REFRACTION THROUGH LENSES 49 

where F^ and F^ represent the same quantities as on 
page 44. 

The nodal points N and W possess another im- 
portant property in addition to that already demon- 
strated. If planes be drawn through these points 
perpendicular to the axis of the lens, an object in 
the first plane will have its image in the second 
plane, and the image and object will be of the same 
size and on the same side of the axis ; in other words, 
the line joining the points where the incident and 
refracted rays meet, respectively, the first and second 
planes, is parallel to the axis. The two planes are 
called the principal planes, and the points where 
they meet the axis are called the principal points. 
In lenses, the principal points and nodal points coin- 
cide ; but this is not so in all optical systems, as we 
shall hereafter learn. 

To prove this property of the principal planes, let 
be the linear magnitude of an object, i its image after 
one refraction, and i^ its image after two refractions. 

Then from equation (1), page 39, we have 



= 0, 



Hence, 



X 

/ 


1_ 


n • i 

f 




i' 

/i 






i 


nf 


and - = 
i' f'fi' 





50 HANDBOOK OE OPTICS 

But in demonstrating the properties of nodal 
points we have seen that AO : A^O = r : r^. Since 
in the first refraction C is conjugate to iV", then AN 
is represented by / and AC by /'; and since in the 
second refraction iV^ is conjugate to (7, then A^ C is 
represented hy f\' and A^H' hy fy Hence, 

Applying formula (a), page 30, to the first refrac- 
tion, we have 

1 n_n — l f'-\-nf__n — l .^^ 

J+J7-=^^; or, -^^^ = -^. w 

Similarly, applying this formula to the second 
refraction. 

Dividing (2) by (1), we derive 

ff(fl±nfA = L = f 



from which 






or, /// + nff, =f,f' + «//i, and /// =f,f'. 



KEFRACTION THROUGH LENSES 



51 



Therefore 



the equation — 

^ 



^^ becomes o = i\ 
J J I 
which shows that object and image are equal when 

they lie in the planes perpendicular to the axis at the 
nodal points of the lens. The significance of the 
principal planes is rendered clear by reference to 
Fig. 17. A thorough understanding of the proper- 
ties of these planes is necessary to the further study 




Fig. 17. 



of refraction. A ray of light RW meets the first 
surface at R , It is directed toward P, but before 
reaching P, it is refracted at R ^ so as to assume the 
direction R R" ; at W it is again refracted so that 
its direction on leaving the lens is R'^ R" . If R^R^^ 
be prolonged backward, it will meet the plane P'W 
at P^ and PN and P'N' will be equal. Any other 
ray directed toward P will after refraction appear to 
come from P' ; in other words^ P' is the image of P. 
It must not be supposed, however, that a real object 
placed in the lens substance at PN would have for 



52 



HANDBOOK OF OPTICS 



its image P'N' ; for such an object, as seen by an 
eye, v/ould have undergone refraction at one surface 
of the lens only. PN and P^N' are both virtual. 

By making use of the properties which belong to 
the principal planes, we can construct the image of 
an object after refraction through a lens. Let PA 
(Fig. 18) be the linear dimension of an object, P 
and P^ the principal foci of the lens, H and W the 



F 


R 


— 


R 


A 


^\.F H 


h'^\f^ 


^^^^v^ 




\\ 


S 


S' 



Fig. 18. 



principal points, RS and ^^aS'' the principal planes. 
From the point P draw a ray PFS ; then since it 
passes through the anterior focus, it must after re- 
fraction be parallel to the axis, and from the prop- 
erty of principal planes just demonstrated US and 
E^S' must be equal. Hence S^P' represents the ray. 
Next, take a ray PP from P, parallel to the axis ; 
then it must after refraction pass through the pos- 
terior focus F^^ and Hit and S'P' are equal. 



REFRACTION THROUGH LENSES 53 

B!FP^ will represent the ray after refraction, and 
P^ the point of intersection of the two rays from P, 
will be the image of P. Similarly, we can show that 
any other point of PA has a corresponding image in 
P'A^ and P^ A! is the image of P^. To find the 
size of the image w^e nse the similar triangles PAP 
and PUS, or P^ A: P^ and H'R'F, When the image 
is real, as in our figure, it lies on the opposite side of 
the axis to PA, and it must be considered negative. 
As on page 35, the distance of the object from the 
first principal focus is denoted by the letter it, and 
the distance of the image from the second focus is 
denoted by ^6^ Hence, we have the following equa- 
tions : 

PA^AP^ ^ ^u 

HS~ PH' ^^^' -i" p' 



., P'A P'A^ P 



By referring to page 38, w^e see that these equa- 
tions are the same as those which we found to deter- 
mine the size of the image after one refraction ; but, 
instead of one principal plane from which all dis- 
tances are measured, we now have two such phanes. 

Having solved the problem for the double convex 
lens, we may, without repeating the investigation, 
apply the same formulae to other lenses by making 
suitable changes in the signs of radii and foci. 



54 HANDBOOK OF OPTICS 

The position of the principal or nodal points may 
be found from the equations, 

cF 

AN (which we call A) = 1 , 

and A'N'iorh')- ""-^^ 



F^ + F^-c 

The focal distance of the lens is found from the 
equation 

If, in these equations, we replace the value of F^ 
and F^ in terms of n and r, we obtain 



h = 

(r + ?•') — (« — l)c 


(1) 


U - "''' 


(2) 


(r + r') — (n— 1)<? 




(3) 



{n — 1) [r + r' — (n — !)<?] 

In spectacle lenses the thickness is so slight, as 
compared with the radius of curvature, that it may 
be ignored. Such lenses are spoken of as thin lenses. 
In thin lenses the nodal points coincide with the 
centre of the lens ; and the equation which deter- 
mines F reduces to tiie form 



REFRACTION THROUGH LENSES 55 



F """' 



(H-l)(r + r') 



This equation serves to determine the focal length 
in all kinds of spectacle lenses. If both r and / 
are positive and equal, we have 



F = 



2(« - 1) ' 



and, if we reckon n as 1.5 for the glass of which 
spectacles are made, we have F = r. If r' is infinite, 
that is, if the lens is plano-convex, then 

n — 1 

when ^ is 1.5. If r and r^ are both negative, the 
lens is bi-concave, and we have 

F = 



(^-l)(r + /) 



Hence the principal focal distances are negative. 
This means that rays which are parallel to the axis, 
meeting the lens, do not intersect the axis after 
refraction ; but they would, if prolonged backward, 
meet the axis on the same side of the lens as that 
from which the ra3^s proceed. Similarly, rays which 
are parallel to the axis after refraction, do not come 
from a real point ; but before refraction they are 



56 



HANDBOOK OF OPTICS 



directed toward a point on the axis and on the 
opposite side of the lens to that from which the light 
proceeds. Thus we see that in the concave lens both 
foci are virtual ; also that the first focus lies behind 
the lens, and the second focus lies in front of the 
lens, considering, as we do, that the side of the lens 





Fig. 19. 



which is turned toward incident light is the front 
of the lens. 

In Fig. 19, (1) represents the action of a convex 
lens, (2) represents the action of a concave lens. 
All rays from 0, the anterior focus of the convex 
lens, are after refraction parallel to the axis ; all 
rays, which before refraction are parallel to the axis, 
meet at the posterior focus 0^ Convex lenses are 



REFRACTION THROUGH LENSES 57 

called collective lenses. In the concave lens rays 
which are parallel to the axis before refraction are 
rendered divergent after refraction. If prolonged 
backward, they would meet at 0\ 0' is then the 
posterior focus of the lens, for it is the virtual inter- 
secting point with the axis of rays wliich before 
refraction are parallel to the axis. Similarly, rays 
converging to the point before refraction are 
parallel to the axis after refraction. is therefore 
the anterior focus of the lens, since it is the virtual 
intersecting point with the axis of rays which are 
parallel to the axis after refraction. Concave lenses 
are called dispersive lenses. 

If, in equation (4), page 47, we disregard the 
thickness of the lens, making c equal to zero, and if 
we substitute F^ the focal distance of the lens, for 
its equivalent, the equation reduces to the form, 

1+1=1. 

f A 

Equation (4) can be reduced to the same form, 
irrespective of the value of (?, but not so simply as 
when c is equal to zero ; and, as we need apply this 
formula to thin lenses only, we shall not make the 
substitution for thick lenses. From the expression 
thus obtained, we see that after refraction by convex 
lenses, conjugate foci lie on opposite sides of the 
lens and are real so long as both conjugate points 



58 HANDBOOK OF OPTICS 

are without the principal foci, but if one point is 
within the principal focus, and real, the other lies 
without the principal focus, on the same side of the 
axis as the first point, and is virtual. We also ob- 
serve from a study of this equation that as a point 
approaches the lens, its conjugate moves in the same 
direction, that is, it recedes from the lens when it is 
real, and approaches the lens when it is virtual. 
Since an object lying in one conjugate plane has its 
image in the other, we may apply the foregoing con- 
clusions to the images of objects. Since, as we have 
seen, real images are inverted, a real and inverted 
image of an object will be formed by a convex lens, 
provided the object be not within the principal focus 
of the lens. If the object be placed at the principal 
focus of the lens, the rays from every point of the 
object Avill be rendered parallel, and no image will 
be formed. If the object be placed within the prin- 
cipal focus, the rays from every point of the object 
will, after refraction, be divergent, and if prolonged 
backward will meet in a virtual focal plane, thus 
forming a virtual image. 

If, in the equation which we have been studying, 
we make F negative, as in the concave lens, we 
shall see that rays from an object will not, after 
refraction, meet in a real focus. 

Since real images are formed by the actual inter- 
sections of rays of light, they may be depicted upon 



REFRACTIOX THROUGH LEXSES 59 

a screen, or upon a sensitive photographic plate 
which is capable of retaining the impression; or, 
again, upon the retina of the eye, a nervous mechan- 
ism through which the impression is transmitted to 
the brain, where it is manifested as vision. Virtual 
images, not being actual intersecting points of rays, 
cannot be so depicted. 

Plano-convex lenses act in the same manner as 
bi-convex lenses; and plano-concave act as bi-con- 
cave lenses. Periscopic lenses act as convex or 
concave lenses according as the convex or concave 
surface has the greater curvature. Reference to 
the equation 



renders this apparent. 

The power of a lens is inversely proportional to 

the focal length. If T represents the focal length, — 

is the power of the lens. Lenses may be numbered 
according to their focal length or according to their 
power. A lens which has a focal length of ten 
inches is called a ten-inch lens. Its power is ex- 
pressed by y^Q-. Since, in lenses which have equal 
curvature at the two surfaces and whose index is 
1.5, the focal length is equal to the radius of curva- 
ture, the power of the lens is expressed by -. 



60 HANDBOOK OF OPTICS 

We obtain the power of two or more lenses used 
in combination by adding the expressions denoting 
the power of each lens ; but this simple method 
applies only when the distance between the centres 
of the lenses is so slight that it may without error 
be neglected. That the power of two or more lenses 
used in combination is equal to the sum of the 
powers of the lenses, may be proved from the rela- 
tion between bi-curved and piano-curved lenses.* 
Since the principal or nodal points and the optical 
centre of a thin lens coincide, any bi-curved thin 
lens is equivalent in all respects to a piano-curved 
lens of the same focal length. Hence if we wish to 
combine two lenses, we may first replace them by 
two piano-curved lenses with their plane faces in 
contact. We then have as the result of this com- 
bination a bi-curved lens ; and the focal length of 
this lens is found from the equation 

F= -^1^2 

F^ + F^-o 

When the thickness of the lens is neglected, this 
becomes 

f, + f; 

F^ being the focal length of the first, and F^ that of 
the second of the component lenses. 



EEFRACTION THROUGH LENSES 61 

From this we find 

F F^ F^ 

To find the power obtamed by combining a lens 
of ten inches focal length with one of twenty inches, 
we have 



1=1- 4- -1 = 1 

F 10 20 20 



-^= tt: + ttt: = 77;^ ; or, J^ = 6| inches. 



In order to avoid fractions in the addition of 
lenses, another method of numbering them is era- 
ployed. This method, which possesses many advan- 
tages, has almost entirely displaced the old method 
in ophthalmology. The unit of power is that of a 
lens whose focal length is one metre. This unit is 
called a dioptre. A lens which has a focal length 
of one-half metre has thejefore a power of two 
dioptres. A lens whose focal length is two metres 
has a power of .5 dioptre, and so on. Hence to find 
the power of any number of lenses used in combina- 
tion, we have only to add their dioptric values. 
Thus a lens of two dioptres in combination with 
one of three dioptres is equivalent to a lens of five 
dioptres. 



CHAPTER IV 

THE EYE AS AN OPTICAL SYSTEM 

The eye as an optical system consists of three 
refracting surfaces and three media. The first sur- 
face is the cornea. Strictly speaking, this is not a 
spherical surface; it conforms more closely to the 
small end of an ellipsoid of three unequal axes, but 
we may without appreciable error replace in our cal- 
culations the normal cornea by a spherical surface. 
The anterior and posterior surfaces of the cornea 
being very nearly parallel, and the refractive index 
of the cornea and aqueous being practically identi- 
cal, the cornea may be disregarded and the aqueous 
humor may be considered the first refractive me- 
dium. After traversing the aqueous, light enters 
the crystalline lens. This is not composed of a 
homogeneous medium, but of numerous layers, the 
density of which increases from the outer to the 
central part of the lens. To trace the path of light 
through each of these layers would be an impossible 
task. Helmholtz, accordingly, divides the lens into 
three portions with increasing index, — the cortical 

62 




THE EYE AS AN OPTICAL SYSTEM 63 

or outer portion, the intermediate, and the nuclear 
portion. From these he has determined the mean 
refractive index of a lens having the same curvature 
and refractive power as the crystalline lens. Refer- 
ence to Fig. 20 will sliow that this does not mean 
that the refractive index of Helmholtz' 
equivalent is equal to the mean of the 
indices of the three portions into which 
the lens is divided. The index of tlie 
equivalent is greater than the greatest 
index of the component portions of tlie 
lens ; for if the index of the entire lens 
were equal to that of the nucleus, its refractive 
power would be less than in tlie lens as constituted. 
This will be seen from the figure. The letter n 
indicates the nuclear portion, which has a small 
radius of curvature ; and therefore, acting alone, 
it would have a greater refractive power than the 
entire lens of the same index, for its curvature is 
greater. The outer portion has the same effect as 
if two divergent menisci were added to the nucleus. 
The index of the outer portion being less than that 
of the nucleus, the addition of the two menisci has 
a less divergent effect than if they had the higher 
index of the nucleus. Thus we see that the refrac- 
tive power of the lens with increasing index is 
greater than if it were composed of homogeneous 
material with the index of the most highly refract- 



64 HANDBOOK OF OPTICS 

ing part of the lens ; and consequently the index of 
a homogeneous lens having the curvature and power 
of the crystalline lens must be greater than that of 
the nucleus of the lens. This equivalent, as deter- 
mined by Helmholtz, is 1.4371. 

Another effect of this increasing index is to 
diminish spherical aberration. We have seen that 
a spherical lens has a greater refractive power for 
rays that meet it at a distance from the axis than for 
those which pass near the axis. By means of the 
physiological arrangement of layers with increasing 
index, those rays which meet the lens near the axis 
are acted upon by the more highly refracting nucleus, 
while those which are remote from the axis escape 
this portion and are refracted only by the cortical 
layers of the lens. 

Finally, after passing through the lens, light enters 
the vitreous, whose refractive index is the same as 
that of the aqueous. 

Gauss, the eminent mathematician, has by his 
researches rendered it possible for us to trace the 
path of light from an external object through the 
media of the eye to its focus on the retina.* Prior 
to his work other mathematicians, notably Moebius, 
had investigated refraction through a number of 
media, but they neglected the thickness of the lenses, 

* Gauss, "Dioptrische Untersuchungen," Werke, Band V. 
Gottingen, 1867. 



THE EYE AS AN OPTICAL SYSTEM 65 

thereby causing an appreciable error in dealing with 
lenses of considerable thickness in comparison with 
their focal length, as in the case of the lens of the 
eye. Furthermore, Gauss demonstrated that an 
optical system of any number of spherical surfaces 
and media, the surfaces all being centred on the 
same axis, has certain points and planes which are 
very useful in determining the optical effect of 
the system. These are called cardinal points and 
planes, and are similar to the cardinal points and 
planes of single lenses. As in single lenses we 
have : 

The first and second princiioal j^oints and the first 
and second principal plane^^ and 

The first and second principal foci and the first 
and second principal focal planes. 

When in any system the positions of these points 
and planes have been determined, the solution of the 
system is complete. 

There are also two other points, the first and 
second nodal points, which are similar to the 
nodal points of a lens. These points, though 
useful, are not necessary to the solution of the 
system. Their properties were first demonstrated 
by Listing. 

Let P (Fig. 21) represent a point on the axis of 
the system ; and let HR represent the first surface 
or cornea of the eye ; W R the anterior surface and 



HANDBOOK OF OPTICS 



o» 



a 




WR' the posterior surface of the 
crystalline lens. A ray of light passing 
through P meets the cornea at i2 ; it 
will be refracted so as to assume the 
direction RR' ^ and if there were no 
further refraction, it would meet the 
axis at Q\ But after travelling the 
distance RR\ it meets the anterior 
surface of the crystalline lens and is 
refracted so as to assume the direc- 
tion R R'' ^ and if it continued in this 
direction, it would meet the axis at 
Q^^ . But at the posterior surface of 
the lens it is again refracted so that 
its final direction is R'^Q, Hence we 
see that in the first refraction Q' is 
conjugate to P ; in the second refrac- 
tion Q^' is conjugate to Q' ; and in 
the third and last refraction Q is con- 
jugate to Q'^ . The point Q being 
the point of intersection of the ray 
with the axis after the three refrac- 
tions, is conjugate to P with reference 
to the entire system. 

To find the relation between the 
position of a point and its conjugate 
after refraction through the eye, we 
make use of the folio win or constants : 



Fig. 21. 



THE EYE AS AN OPTICAL SYSTEM 67 

The refractive index of air, which is denoted by 1 

The refractive index of the cornea and aqueous n 

The refractive index of the crystalline lens . . n' 

The refractive index of the vitreous . . . . n 

The radius of curvature of the cornea . . . r 
The radius of curvature of the anterior surface 

of the lens r' 

The radius of curvature of the posterior surface 

of the lens r'' 

The distance of anterior surface of the cornea 

from the anterior surface of the lens , . . nt^ 

The thickness of the crj^stalline lens , . . , n't^ 

All distances measured from left to right are posi- 
tive, and those measured from right to left are nega- 
tive. In accordance with this convention of signs, 
we use formula (5), page 31. Then r and r' are 
positive and ?^'^ is negative. The distance from H 
to S^ (Fig. 21) is denoted by 7it^, and that from R' 
to S'^ by n^t^. Let the distance of P from the hrst 
surface be denoted by/; and the distance of Q' from 
the same surface by nfy The distance of the second 
image Q^' from the second surface W is denoted by 
n^f^, and the distance of the third image Q from the 
third surface H^^ by nf^.^ 

* The reason for this notation will become apparent in the 
course of the demonstration. 



68 HANDBOOK OF OPTICS 

Hence 

PH=f; HQ>=nf^; H' Q" = n%; and JI"Q = nf,. 

Then at the first refraction, applying formula (6), 
page 31, we have 

1 n 1 — n 



f Vi r 

At the second refraction, 



(1) 



n n' n — n' 



H'Q' n% r 

But H' Q' = HQ' - HH' = nf^ - nt^ ; hence 
1 _ 1 _ n — n' 

At the third refraction, 



(2) 



n/ n n^ — n 



But H'^ Q^' =-- H' Q^' - WW^ = ^7*2 - nH^ ; there 

fore 

1 1 n^ — n 



For convenience we make 

—;r- = ^o' -^ = h' and -—^^h 



(3) 



THE EYE AS AN OPTICAL SYSTEM 69 

From (1) we have 

From (2), /^ = f ^ + —^. 

h + j 

From (3), f^ = t, + ^-p- 



Substituting these values of /j and f^, we have 
1 



^=h, + ^— (4) 

^1 + 






K + \ 



/ 



This is an equation expressing the relation be- 
tween / and /g, from which we can find the conju- 
gate focus of a point, P, in any case. The same 
method may be applied to any number of surfaces, 
but as the equation becomes much more cumbersome 
with each additional surface, it is convenient to ob- 
tain from the known properties of such an expres- 
sion a simpler relation between / and /g. The 
second member of equation (4) is called a continuous 



70 HANDBOOK OF OPTICS 

fraction. If we should neglect all the terms to the 
right of Z^Q, the expression would become 



Kq 



If we neglect all except k^ + — , we have 

^1 



(a) 



^0^1 + 1 



h 



(J) 



Neglecting now all after k^ + , we have 

Continuing this process, we finally embrace all the 
terms and get the true value of the second member 
of the equation. The expressions (a), (5), and (^) 
are called convergents. If we examine the conver- 
gent ((?), we shall see that its numerator is obtained 
by multiplying the new letter k^ by the numerator 
of the preceding convergent and adding the numera- 
tor of the second preceding convergent ; and that 
the denominator is obtained by multiplying the new 
letter k^ by the denominator of the preceding con- 
vergent and adding the denominator of the second 
preceding convergent. This relation is true for all 
subsequent terms of a continuous fraction such as 
the second member of equation (4); we can there- 











^i(Vi + l) + '^o 




k^h + 1 




t^\k^(k,t^+ 1) + h\ + k,t^ + 


1 


t^ik^t^ + l) + t^ 





THE EYE AS AN OPTICAL SYSTEM 71 

fore write out the value of this fraction without 
further calculation. The successive convergents, as 
thus obtained, are : 

(a) 

(^) 

(0 

(Jl 
(A) 

^2[^2l^l(Vl + l) + ^0<+Vl + l] + /i:i(Vi + l) + /i;o (^ 

k^\uxhh+'^)+h\+hh+'^ '(0" 

For convenience we call the last two of these con- 
vergents "^ and - ; then the next expression, which 
will be the true value of the fraction, may be written 

--:-, — f-, and equation (4) becomes 
f^l + h 

1 ^f^k + g 

f U + h- 

If we multiply numerator and denominator of the 
second member of this equation by n^ we shall have 

1 _ nf^k + ng 
or nf^'l + nh =f - iif^ • ^ +/ • ng. (5) 



72 HANDBOOK OF OPTICS 

Since nf^ is the distance of the last image, Q\ from 
the last surface, equation (5) expresses the relation 
between P and its conjugate Q, Tliis is a general 
equation, true for all positions of P ; we may there- 
fore, by making 7if^ and / respectively equal to in- 
finity, find the positions of the principal foci. 

Equation (5) may be written 

(^fk - V)nf^ = nh -f^ng, 



or 






fk 


-1 = 


• 




When 


^/3 = 


= GO, 




f- 


I 




Similarly 


we 


find when / = 


■■ 00 


9 










nfs = 


— 


ng 

7. ■ 



These values determine the first and second prin- 
cipal foci as measured respectively from the anterior 
surface of the cornea and the posterior surface of the 
crystalline lens ; but we cannot construct the image 
of an object until we find the position of the prin- 
cipal planes. These planes are, as we know, conju- 
gate focal planes such that an object in one of them 
will have its image in tlie other, the object and 
image being equal and on the same side of the axis. 
Hence to find these planes in a system of refracting 



THE EYE AS AN OPTICAL SYSTEM 



73 



surfaces, it is evident that we may use equation (5), 
which expresses the relation between conjugate foci. 
If, in this equation, we impose the condition that 
object and image be of the same size and sign, the 
resulting values of / and nf^ will be the distances of 
the first and second principal planes from the ante- 
rior surface of the cornea and posterior surface of 
the crystalline lens, respectively. 




Fig. 22 



In Fig. 22 let PR represent an imaginary ray 
directed toward E, After refraction throusfh the 
system it appears to pass through E^ ; then E and 
E' are conjugate foci. Let F mark the position of 
the anterior, and F the position of the posterior 
focus of the system ; then 



k 



and H^^F' = - '^. 

k 



74 HANDBOOK OF OPTICS 

Since U is tlie point of intersection of the entering 
ray with the axis, IIE=f; and likewise U\ being 
the point of intersection of the emergent ray with 
the axis, H'^U^ is equal to nf^. Equation (5) may 
be written, 

or, 

But by a property of continuous fractions, gl — hk 
= 1.* 

„_,(/-l)(„/.-(^).-|. (6) 

We have also (Fig. 22) FE = HE + HF, but HF 

is in the figure negative ; if it were positive, we 

should have FE = HE - HF ; or, FE^f--. 

k 

Similarly, FE' = nf^ - ^^^^- 

Let the distance FU be denoted by u. As in the 
case of single lenses, it will be most convenient to 
consider u positive when U lies to left of F, and 
negative when it lies to the right of F, as in our 

* Reference to the simpler convergents on page 71 will make 
this apparent. 



THE EYE AS AN OPTICAL SYSTEM 



figure.* Likewise if the distance F'E' be denoted 
by u\ then u^ will be positive when JE' lies to the 
right of F\ and negative when it lies to the left of 
i^^ as in the figure. Then 

f — - becomes — u^ and iifo — ^ ^ becomes 

k k 

and equation (6) thus becomes 



w 



n 






This equation is true for any two conjugate points 
E and E' : and to find the values of u and u^ which 







R 


/ 


Rl- 


r 


R" 


p" - 


____J^ -;:^r^r= 


^^^ 


r 


H' 




H" 




P' «'>'P 




V 


\ 


y 








Fig. 23. 









give to E and E' the properties of principal points, 
we make use of Helmholtz' formula for obtaining 
the relation between the size of an object and its 
image. In Fig. 23 let JI. H', and H" represent the 
points of intersection of the refracting surfaces with 
the axis ; then RH, R'H\ and R^'H" are the princi- 
pal planes of these surfaces. A ray of light passing 

* See Chap. II., p. 35. 



76 HANDBOOK OF OPTICS 

through P meets the principal plane RH at B ; it is 
refracted to R' ^ where it is again refracted to R^ . 
RPH^ the angle which the incident ray makes with 
the axis, is called a, and the angle RP^H^ which the 
refracted ray makes with the axis, is called a^ Simi- 
larly, ^2 and a^ are the angles which the ray makes 
with the axis after the second and third refractions. 
Let &, 6j, h^, represent the distances from the axis at 
which tlie ray meets the successive principal planes. 
As in our previous demonstrations, the refractive in- 
dex of air is unity; that of the aqueous and vitreous 
are each n, that of the crystalline lens n^ ; the dis- 
tance HH^ is nt^, and H'H^' is n^t^. From Fig. 23 
we obtain the following equations : 

HP = -A_ ; HP^ = — ^ ; WP^' = -^. 

tan a tan a-^ tan a^ 

Since in the first refraction IIP and RP^ are con- 
jugate foci, and both negative, we have formula (5), 
page 31, 

1 , n 1— n tan a , n , tan a. -, 



HP HP' r h h 

from which n • tan a^ = tan a + kj). 

Referring again to Fig. 23, Ave see that 

h-^ = l + nt^* tan a^ 



THE EYE AS AN OPTICAL SYSTEM 77 

In exactly the same way it is proved that 

7\! tan a^ = n ' tan a-^ + h-J)-^^ and 52 = ^1 + ^^^2 * ^^^^ ^2' 

and so on. By these equations all the quantities 
n • tan a-^, 5^, n' tan a^, h^^ ii • tan ag, may be expressed 
in terms of tan a and h. Their values become 

n tan a^ = k^b + tan a, 

?>^ = O^Vi + 1)^ + ^1 tan a, 
n' tan a^ = {k-^i^k^t-^ + 1) +A:q^ J + (/qf-^ + 1) tan a, 

and so on. 

The coefficients of b and tan a in these equations 
are seen to be respectively the numerators and de- 
nominators of the successive convergents on page 71. 

Hence we may write out the value of b^^ thus : 
b^ = gb + h tan a ; and n • tan a^ = kb + 1 - tan a, Avhere 
g^ A, k^ and I have the same significance as on page 71. 

If represent the linear dimension of an object 
and i^, 2*2, and i^ the corresponding dimensions of the 
successive images, then from Helmholtz' formula, 
page 39, we shall have 

• tan a = n 'i-^' tan a-^ = n^ - i^ • tan a^ = n - i^- tan a^. 

The relation between the object and its final 

image is 

• tan a = i^' n ' tan a^, (7) 



78 HANDBOOK OF OPTICS 

But n ' tan a^ = kb + I - tan a, and from tlie figure, 
b = HP tan a, or since HP = — /, b = —f • tan a. 
Substituting this value, we have 

n . tan a^= I - tan a — kf * tan a, 

or n . tan a^ = k * tan a ( / j, 

and equation (7) becomes 

• tan a = i^k • tan al /). 

Referring to page 75, we see that 

Therefore o = i^ku^ and if object and image are 
equal, which is the condition to be imposed in order 
that u and u^ may determine the principal points, 

= L and u = -* 
From the equation 

uw = -, we lind ii^= — — . 

k^ k 

Thus we have determined the values of u and u^ 
which represent the distances of the first and second 
principal points from the first and second principal 
foci, respectively ; and since these distances are the 



THE EYE AS AN OPTICAL SYSTEM 



79 



first and second principal focal distances, we have 
only to find the numerical value of k in order that 
the solution of the system be complete. The value 
of k can be obtained if we know the radii of the 
refracting surfaces, the distances between these sur- 
faces, and the refractive indices of the media. In 
the human eye these quantities have all been meas- 



p R _ R 

a| ^^-^F E E^ ^^F^ * ^' 

^\ — ^P' 

S S' ^ 



Fig. 24. 



ured by careful and scientific investigators. Of 
these the name of Helmholtz is most conspicuous. 

Before substituting these values it will be well to 
show the construction of the image after refraction 
through a number of media, and to demonstrate the 
properties of the nodal points. The geometrical 
construction of tlie image is almost a repetition of 
that for sinoie lenses. 

In Fig. 24, RS and R' S' represent the principal 
planes, F and F' the principal foci, AP the linear 



80 HANDBOOK OF OPTICS 

dimension of the object, and A^P^ that of the image. 

The anterior focal distance UF is denoted bj^ F^ and 

the posterior focal distance F^F' by F', If, as in 

our former notation, u denote the distance of the 

point A from the anterior focus, and u' the distance 

of its image from the posterior focus, then FA = u 

and F^A' = u'. From the similar triangles PAF and 

EFS we have 

PA^FA 

ES EF' 
andas^AS^^^'P', ^^ ^^ 



P^Al EF 

Since in our construction the object and image 
are on opposite sides of the axis, the image ^ must 
be considered negative, and we have this equation 
to determine the size of the image : 

on. F 
; = — . or ^ = — — 

— I F u 

Also from the triangles E^F^R and F'A^P^, we 
obtain the relation 

F' . u^ 

— I u^ F' 

This construction differs from that for single 
lenses only in the respect that in lenses F and F' 
are equal, while in the present case F is to F' as 



THE EYE AS AN OPTICAL SYSTEM 



81 



1 to ??, that is, as the index of air is to that of 
the final medium or vitreous. If, after refraction 
through any number of media, light enters a medium 
of the same index as that of the first medium, the 
system acts toward light as a single lens; if the 
final medium has not the same index as the first, 
the system is analogous to a single refracting surface. 







S 




S' J 


p 


F/^ 


E 


N 


E 


nV^ 


F 


/ 




^ 




^^ 






R 


x'' 




R 












P/^ 













Fig. 25. 



We have seen that in refraction at one surface 
there is one nodal point at the centre of curvature, 
and in refraction through lenses there are two nodal 
points. The characteristic property of nodal points, 
we remember, is that a ray directed toward one of 
them appears after refraction to come from the other, 
and in a direction parallel to that before refraction. 
Let us then take a ray PR (Fig. 25) directed 
toward the point N on the axis, so that after refrac- 
tion through the system it appears to pass through 



82 



HANDBOOK OF OPTICS 



N^ m a direction WP^ parallel to PR ; then N and 
N^ are nodal points. To find the position of these 
points we construct the principal planes RS and 
R^ S^ and the principal focal plane F^P^ . The ray 
PR meets the first principal plane at R, It must 
after refraction meet the second principal plane at 
R so that ER = E' R' . P' is its point of intersec- 
tion with the focal plane P'F' , If we draw any 
two rays from P^, passing through the system, they 



p 




■^N 


EN-^ 


\.F' 




A 


^\.F^^^^^^ 


^' 









^^^^^~-^:::^ 


r^' 








P 



Fig. 26. 



must after refraction be parallel. Let P'N^ repre- 
sent one of these rays, and P^S' ^ parallel to the axis, 
the other. P' S' must after refraction pass through 
the anterior focus F, SF will represent this ray 
and RP will represent the ray P^ N' after refraction. 
FN^ the distance from the first focus to the first 
nodal point, is FF + FK The triangle RFJV is 
equal to R E' N\ and FES is equal to WF'Ri 
hence FE + EN =^ WN' + N' F\ or FN= F' , In 
the same way we find N' F' = F, When, as in lenses, 



THE EYE AS AN OPTICAL SYSTEM 83 

F and F' are equal, it is clear that the nodal and 
principal points coincide. It is also apparent that 
the distance between nodal points is the same as 
that between principal points. The size of the 
image may be determined if we know the distance 
of object and image from the first and second nodal 
points respectively. In Fig. 26 AP=o^ and A^P' = i ; 
then from the similar triangles J-PiV^and A'P'N'^ 

AP AN AN 



TWr ^^ 



A^P^ A^N^ -i A^N' 

The following is a table of measurements of the 
constants which we shall require in the determina- 
tion of the refractive power of the eye : 

Radius of curvature of the cornea (r) . 7.829 mm. 

Radius of curvature of the anterior 
surface of the lens in a state of 
rest (r^) 10 mm. 

Radius of curvature of the posterior 

face of the lens (/') 6 mm. 

Distance of the anterior surface of the 
cornea from the anterior pole of the 
lens (nt^ 3.6 mm. 

Thickness of the lens (n^t^) .... 3.6 mm. 

Index of refraction of tlie cornea, aque- 
ous and vitreous (ji) 1.3365 mm. 

Equivalent index of refraction of the 

crystalline lens Qi'^ 1.4371 mm. 



84 HANDBOOK OF OPTICS 

Substituting these values in the successive conver- 
gents on page 71, we deduce the following : 



^^0 = -"-^ =-.043 



L = — = 2.6936 

n 



t.=^^ = 2.505 
^ n' 



r 



]c^=:VLz±^- .QIQS 



yto^j + l=.8842, ^1^1 + 1 = .9731, 

^i(Vi + 1) + ^0 = - -0518, 
hWihh + 1) + '^os + Kh + 1 = ^ = -7544, 
t^ik^t^ + 1) + ii = A = 5.1312, 

KiklhiKh + 1) + K\ + Vi + 1] 

+ k^ih^f,^ + l) + A;o = A; = -. 0645, 
K\hihh + 1) + i^iS + hh + 1 = Z = .8869. 



THE EYE AS AN OPTICAL SYSTEM 85 

The first principal focal distance EF^ which is 

equal to -, thus becomes 
k 

-15.5038 mm.; and RF or -=- 13.7504 mm. 

k 

Therefore FH, which is equal to FF — HF, is 
— 1.7534 mm.; thus the cornea lies 1.7534 mm. in 
front of the first principal point. The second focal 

distance E^ F\ being represented by — -, is 

rC 

20.721mm.; and H" F' or -^ is 15.6826 mm. 

Therefore F\ the second principal point, lies 5.0884 
mm. in front of the posterior surface of tlie lens, 
and since this surface is 7.2 mm. behind the anterior 
surface of the cornea, the second principal point lies 
2.1116 mm. behind the cornea. The distance be- 
tween the two principal points is .3582 mm. The 
position of the first nodal point of the eye is, as we 
know, found by laying off the distance FN equal to 
E' F\ the second focal distance ; as thus found, N 
lies 6.9706 mm. behind the anterior surface of the 
cornea ; and since the distance between nodal points 
is the same as that between principal points, the sec- 
ond nodal point is 7.3288 mm. behind the anterior 
surface of the cornea. With these points determined 
our knowledge of the eye as an optical system is 



86 HANDBOOK OF OPTICS 

complete. The following table giving the positions 
of these points will be found useful for reference : 

Distance of the anterior surface of the . 

cornea from the first principal point 1.7534 mm. 
Distance of the anterior surface of the 

cornea from the second principal 

point .......... 2.1116 mm. 

Distance of the anterior surface of the 

cornea from the first nodal point . 6.9706 mm. 
Distance of the anterior surface of the 

cornea from the second nodal point 7.3288 mm. 
Distance of the first principal focus 

from the anterior surface of the 

cornea 13.7504 mm. 

Distance of the second principal focus 

from the anterior surface of the 

cornea 22.8326 mm. 

The hypothetical eye which possesses these car- 
dinal points is called the schematic eye. 

We have seen that the method of determining the 
size and position of the image is the same in refrac- 
tion at one surface as in refraction at any number of 
surfaces, with the exception that there are two prin- 
cipal and two nodal points in the latter case and only 
one principal point in the former, with one nodal 
point at the centre of curvature. We also know 
that the size of the image is proportional to the 



THE EYE AS AN OPTICAL SYSTEM 87 

anterior focal distance ; and that this distance is 
measured from the first principal point, while the 
second focal distance, which determines the position 
of the image, is measured from the second principal 
point. In tlie eye the two principal points are so 
near to each other that they may with very slight 
error be merged into one. In this way the optical 
effect of the eye may be represented by one refract- 
ing surface whose summit coincides with tlie merged 
principal points. Listing first proposed this simpli- 
fication, and called tlie resulting equivalent the re- 
duced eye.* In this reduction the single principal 
point is between the two principal points of the 
schematic eye as determined by Listing. f The 
ratio of F to F^ remains unchanged, and as F is to 
F' as 1 to n^ we can determine what value n must 
have in the reduced e5^e. Listing takes as the an- 
terior focal distance 15.036 mm., and as the second 
focal distance 20.133 mm., from which we find n is 
equal to 1.3365, and this, as a reference to the table 
on page 83 will show, is the index of the vitreous. 

From the equation F = -, we can determine the 

n — 1 

^ Dioptrik cles Auges, Wagner's " Handworterbuch der Phy- 
siologie." 

t Listing's schematic eye differs slightly from that given in the 
text, since he takes the radius of curvature of the cornea as 8 mm., 
and the distance from cornea to lens and the thickness of the lens 
as each equal to 4 mm. 



88 



HANDBOOK OF OPTICS 



radius of curvature of the reduced eye. Using 
Listing's measurements, this is 5.1248 mm. Figure 
27 illustrates in (1) the schematic eye and in (2) 
the reduced eye. 




Fig. 27. 



Bonders has furnished a less accurate but a more 
useful reduced eye than that of Listing. He neg- 
lects fractions, making the anterior focal distance 
15 mm., the posterior focal distance 20 mm., and the 



radius of curvature 5 mm. From F = 



we find 



n — 1 
n = 1.383, which is the index of water. Calculations 



THE EYE AS AN OPTICAL SYSTEM 89 

as to the size of images with this eye are extremely 
simple. If it be remembered that the size of the 
image is proportiomil to the anterior focal distance, 
and that in Bonders' eye this is 15 mm., while in 
the schematic eye it is 15.5038 mm., it will be ap- 
parent that the image obtained from calculation with 
Bonders' eye is to the actual image in a normal eye 
as 15 to 15.5038. 

Based upon these data, artificial eyes have been 
constructed for the study of tiie refraction of the 
eye. 



CHAPTER V 

THE DETERMmATION OF THE CAEDINAL POINTS OF 
THE EYE IN COMBINATION WITH A LENS 

We must now carry our investigations one step 
further in order to appreciate tlie effect upon vision 
of placing a lens in front of tlie eye. A lens has two 
refracting surfaces, and the eye three such surfaces ; 
hence, if we wish to use the method given in the 
preceding chapter, we must write out the convergents 
for two additional surfaces ; but simpler than this 
would be an independent geometrical construction, 
from which we could find the effect of combining 
two optical systems. But for our purpose it suffices 
to make use of the reduced eye of one surface, since 
we do not care to know the exact size of the retinal 
image. What we wish to know is the relative size 
of images with lenses and without them ; and the 
investigation with the reduced eje furnishes this 
information. 

In Fig. 28 let A represent the centre of a spheri- 
cal lens ; and since we neglect the thickness of the 
lens, the letter A also marks the position of the 
merged principal points of the lens. Let A^ repre- 

90 



THE EYE IN COMBINATION WITH A LENS 91 

sent the position of the principal point of the reduced 
eye. We know that in the lens A the proportion of 
curvature at the two faces is immaterial, in other 
words, any thin bi-spherical lens may be replaced by 
a piano-spherical lens of the same focal length ; and 
for the sake of simplicity we shall make this substi- 
tution. To find the cardinal points of the combina- 
tion of eye and lens, we use the formulae deduced in 
the preceding chapter. If F denote the anterior and 



F, 




Fig. 28. 
F^ the posterior focal length of the combination, then 

1 71 » 

F= -^ and F' = , in Avhich n represents the index 

k k 

of the reduced eye, that is, if we use Listing's re- 
duced eye, it is equal to the index of the vitreous, 
and if we use Bonders' reduction, n is equal to the 
index of water. The thickness of the lens, which 
in our formula is represented by nt^^ is equal to zero ; 
and the distance of the lens from the eye, which is 
represented by n^t^-, becomes t^, since n'^ being now 
the index of the air, is unity. This distance between 
the eye and lens may be conveniently denoted by the 
letter e ; then t^ in the formula is replaced by So 



92 HANDBOOK OF OPTICS 

Thus we have h^ = , in which r represents the 

r 

radius of curvature of the first surface of the lens and 
n the index of the lens ; in the piano-spherical lens 

is equal to the focal length of the lens. If we 



1—n 



denote this by F-^^ we have k^^ — —• The radius of 
curvature of the second face of the lens is infinite, 
hence ^j = 0. In the same way we find k^ = — , 

F^ being the anterior focal length of the eye. Mak- 
ing these substitutions in the expressions (^), (Jc), 
and (0» page 71, we derive 

e + F. 
._e_±F^±F^ 

Hence, F = \ = —^^ ; 

Wi^ n^ nF,F,^ ^ F,F,< 

k F^ + F^ + e F^ + F^^ + e 

With the convention of signs whicli we are using, 
F^ and F^ , being measured to the left of A and A' 



THE EYE IN COMBINATION WITH A LENS 93 

respectively, are both negative. In order that the 
formulae may be applicable in any case without pre- 
fixing the minus sign to F^ and F^, we must change 
the sign of these quantities. Making this alteration, 
the equations become 

jT^l^ F,F, . ^,_ n^ F,F4 

h F^ + F^-e' h F^ + F,^-e 

It is in this form that these equations are usually 
written. To find the distance from the lens to the 
anterior focus F^ we have 

AF-^- F,{F,-e^ , 

k F^ + F^-e' 

and to find the distance from the principal point A^ 
of the eye to the posterior focus F , we have 

k F^ + F,^-e 

Having found the positions of the two foci and the 
two focal distances, we know also the positions of the 
two principal points, and the solution of the system 
is complete. 



CHAPTER VI 

ERRORS OF REFRACTION — LENSES USED AS 
SPECTACLES 

We have studied the eye as an optical system, 
taking as our measurements those found to exist 
with close approximation to uniformity in a large 
number of eyes. The posterior focal distance of 
this system we have found to lie 22.8326 mm. behind 
the anterior surface of the cornea. When the retina 
lies at the same distance from the cornea, the image 
of a distant object will be accurately formed on the 
retina. When the retina lies in front of the focus 
of the eye, the image of a distant object will be 
blurred. This condition of the eye is called hyper- 
opia ; and when the retina lies behind the focus, the 
resulting condition is called myopia. 

Eyes so constituted that retina and principal focus 
do not coincide are said to be ametropic, or affected 
with errors of refraction. When retina and focus 
coincide, as in the normal eye, the condition is called 
emmetropia. A distant object will be clearly seen 
by a healthy emmetropic eye, but the image of a 
near object will fall behind the retina. In order 

94 



ERRORS OF REFRACTION 95 

that a near object be clearly seen, either the eye 
must be elongated or the focus must be brought 
forward. The latter change is the one which occurs 
by an increase in curvature of the crj^stalline lens 
(principally of the anterior surface of the lens) 
uncier the influence of the ciliary muscle. This 
change in curvature is called accommodation, ^j it 
we are enabled to adapt the eye for varjdng distances. 

Since the curvature of the lens is increased during 
accommodation, the optical system of the eye in this 
state differs from that when the eye is adapted for 
a distant object ; the focal distance has been sliort- 
ened hj the exercise of accommodation, and the retina 
now lies behind the principal focus ; in other words, 
the eye has become myopic. Thus we see that the 
emmetropic eye can render itself myopic in order 
to see near objects. Similarly, a hyperopic eye, 
possessing sufficient accommodative power, may be- 
come emmetropic to see distant objects, and by a 
still further increase in curvature of the lens may 
even become m3'opic and thus see clearly near objects.* 
The myopic eye is unable to increase its focal dis- 
tance, thus bringing the principal focus back to the 
retina ; it cannot therefore see distant objects clearly. 

Since an eye may be myopic either from increase 
in curvature or from increase in the antero-posterior 

* These conditions, however, are not inchided in the usual 
acceptation of the words emmetropia and myopia. 



96 HANDBOOK OE OPTICS 

diameter, we have curvature myopia and axial 
myopia. Similarly, we have curvature hyperopia 
and axial hyperopia. In axial hyperopia and myopia 
the eye as an optical system is normal, the defect 
being in the position of the retina; in curvature 
ametropia the deviation from the normal is in the 
optical system.* 

Without using the formulae for the cardinal points 
of the eye and lens in combination, we can easily 



Q 



Fig. 29. 

understand how lenses can in hyperopia and myopia 
bring the retina and principal focus into coincidence. 
Let A' R (t^ig- 29) represent the cornea of a hyper- 
opic eye whose principal focus is at F^ behind the 
retina. Rays of light parallel to the axis before 
entering the eye will after refraction meet at Fc^^, 
In order to bring these rays to a focus at F on the 
retina, we introduce a convex lens A. Let Q be 
the principal focus of this lens ; then a ray PR 
parallel to the axis will, after passing through the 

* hidex ametropia, which in optical effect resembles curvature 
ametropia, occurs in exceptional cases. 



ERRORS OF REFRACTION 97 

lens, take the direction RQ ; but at R' it is refracted 
by the eye and assumes the direction R'F. 

Let A^R C^ig- 3^) i^epresent the cornea of a 
myopic eye wliose focus F^ lies in front of the 
retina ; tlien rays of light parallel to the axis before 
entering the eye will, after refraction, meet at F^. 
A ray passing through the point Q will meet the 
axis at its conjugate focus behind the principal focus. 
Let Q be so taken that its conjugate lies on the 
retina; then Q is the far point of the eye, since light 





Fig. 30. 

from any more distant point will meet the axis in 
front of the retina. If we place at A a concave 
lens whose focal length is A Q, then any ray paral- 
lel to the axis will, after refraction through the lens, 
appear to pass through Q, and its direction will be 
QR . But the ray QR will, after refraction by the 
eye, meet the axis at F, Hence a concave lens 
whose focal length \^ AQ will bring parallel rays 
to a focus on the retina. 

It is clear from these diagrams that the farther 
from the eye the convex lens is placed the weaker 



98 HANDBOOK OF OPTICS 

is the lens required to bring the retina and focus 
into coincidence, since AQ is the focal length of the 
required lens. In the case of myopia the opposite 
is true, for AQ^ the focal length of the lens, dimin- 
ishes as the lens is removed from the eye. 

We can determine the amount of shortening of 
the eye in hyperopia, or of lengthening in myopia, 
from the equation 

f f 

In this equation F and F^ are the focal distances 
of the eye, and / and /' are the conjugate distances 
^^^and^^^. 

This equation is true not only for refraction at 
one surface and for lenses, as we have seen; it applies 
equally to a system of refracting surfaces. That 
this is so, follows readily from the equation 



/ 1 n 

uu' = • -9 



k k 



1 



n 



since u^f-F, u^ =f - F\ and -.t.'-^^FF^. 

k k 

Applying this equation in myopia, if Q^ the far 
point of the eye, is 100 mm. from the first principal 
point of the eye, then / is 100 mm. Substituting for 
F^ the anterior focal distance, its value 15.5038 mm., 
and for F^ its value 20.721 mm., we find the corre- 



ERRORS or REFRACTION 99 

sponding value of/' to be 24.5219 mm. That is, 
the retina lies 24.5219 mm. behind the second prin- 
cipal point of the eye. Subtracting 20.721mm., 
which is the distance of the retina from the second 
principal point in the normal eye, we have 3.8009 mm. 
as the amount of lengthening. 

In hyperopia if rays directed toward Q (Fig. 29) 
would be focused on the retina, and if the distance 
of Q from the first principal point of the eye is 
100 mm., tlien / in our equation becomes — 100 mm. 
Making the proper substitutions as before, we find 
/' to be 17.9402 mm., and the amount of shortening 
is 2.7808 mm. From this we see that the shortening 
in hj'peropia is less than the lengthening in the same 
degree of mj^opia. 

Since the strength of the lens required to correct 
an error of refraction varies with the position of the 
lens, it is clear that we cannot measure the error 
by the correcting lens unless we adopt some fixed 
position at which the lens should be placed. To be 
strictly correct, this should be at the first principal 
point, from which Q is measured, but as this is im- 
possible, the position at which spectacle lenses are 
ordinarily worn, about 15 mm., in front of the cornea, 
is taken as the standard. In our example of the 
myopic eye, 100 mm., as measured from the first 
principal point E (Fig. 22) indicates the position of 
the far point in a myopia of 10 dioptres; but the 



100 HANDBOOK OF OPTICS 

focal length of the correcting lens, placed 15 mm. in 
front of the cornea and consequently nearly 17 mm. 
in front of E^ must be a trifle more than 83 mm. 
Taking this as 83 mm., the dioptric power of the 
lens is slightly more than 12 D. In tlie second ex- 
ample we have a hyperopia of 10 dioptres as meas- 
ured from E^ while the degree, as indicated by the 
correcting glass placed 15 mm. in front of the 
cornea, is about 8.5 D. 

If the crystalline lens has been extracted from the 
eye, there is only one refracting surface and one 
medium, for the index of the aqueous and vitreous 
are identical. The second principal focal distance 
of the aphakic eye is therefore derived from the 
equation 

nr 



F^ = 



n — \ 



and from this the focus is found to lie 31.095 mm. 
behind the anterior surface of the cornea. The diop- 
tric power of the aphakic eye is therefore 32.16 D. 
The principal focus of the normal eye lies 22.8326 mm. 
behind the anterior surface of the cornea ; hence its 
dioptric power is 43.8 D. Subtracting the power 
of the aphakic from that of the normal eye, we find 
that the action of the crystalline lens, when adapted 
for distant vision, is nearly equivalent to that of a 
lens of 11.5 D. placed in contact with the cornea. 



ERROKS OF REFRACTION IQl 

After extraction of the lens from an emmetropic eye, 
a convex lens of 11.5 D. would be required to bring 
the image of a distant object to an accurate focus on 
the retina, if the lens were worn in contact with the 
cornea ; but as this is impracticable, a weaker lens 
is required. If the distance of the lens from the 
cornea be 15 mm., then its focal length must be 
87 mm. (the focal length of a lens of 11.5 D.) 
+ 15 mm. A lens having this focal length repre- 
sents a dioptric power slightly in excess of 10 D. 
Hence after extraction of the lens from an emme- 
tropic eye, we should expect a lens of 10 D. to rec- 
tify the refractive condition for distant objects, and 
this corresponds very closely with what we find in 
practice. 

Let us now examine the condition which exists 
after extraction of the lens from a mj^opic eye. If 
the myopia is caused by excessive curvature of the 
cornea, the amount of myopia relieved by the ex- 
traction is the same as in emmetropia, that is, if 
there existed prior to extraction 11.5 D. of myopia 
as measured from the cornea, the eye would be 
rendered emmetropic by the extraction. This, 
however, is not so when, as is usually the case, the 
myopia is due to lengthening of the antero-posterior 
diameter of the eye. As previously shown, the pos- 
terior focus of the aphakic eye of normal curvature 
lies 31.095 mm. behind the cornea. Hence if an eye 



102 HANDBOOK OF OPTICS 

is emmetropic after extraction of the lens, the retina 
also must lie at this distance from the cornea. To 
find what degree of myopia exists in an eye whose 
retina lies at this distance from the cornea, we must 
use the equation 

/ /' 

As we can find from this equation the amount of 
lengthening, if we know the distance (/) of the far 
point from the eye ; so if we know the distance (/^) 
of the retina from the second principal point, we can 
find the distance (/) of the far point from the eye, 
and this distance measures the myopia. Our result 
will be sufficiently accurate if we consider F as 
16| mm., F^ as 21 mm., and /' as 29 mm. We find 
this last value by subtracting 2.1116 mm., the dis- 
tance of the second principal point from the cornea, 
from 31.095 mm., the distance of the retina from 
the cornea, the result being the distance (/') of 
the retina from the second principal point. Making 
these substitutions, we find the corresponding value 
of / to be 66.2 mm. This is the distance of the far 
point of the eye from the first principal point; its 
distance from the cornea is therefore approximately 
51.4 mm. ; or, expressed in dioptres, the myopia, as 
measured from the cornea, is 18.3 D. 

To find what is the required power of the correcting 



ERRORS OF REERACTION 103 

lens for this amount of myopia we shall consider the 
lens placed not, as before, 15 mm. from the cornea, 
but 15 mm. from the first principal point, or about 13 
mm. from the cornea. The position at which spec- 
tacle glasses are usually worn is more remote from 
the eye than as indicated in the standard (15 mm.) 
which we have taken, but this is not so when very 
strong concave lenses are worn. These lenses, when 
tolerated, are worn very near the eye ; hence we 
make the change in distance so as to be more in 
accord with the actual conditions with which we 
meet. It will be noticed that a slight change in 
position of these strong lenses makes perceptible 
change in dioptric power. The focal length of the 
correcting lens in the case which we are considering 
thus becomes 56.2 mm. — 15 mm., or 41.2 mm. The 
power of the lens is accordingly 24 D. From this 
we see that the eye must have a myopia of 24 D. as 
measured by its correcting lens in order to become 
emmetropic after extraction of the lens from the eye. 
Let us now deduce from calculation the lens re- 
quired for distant vision in an aphakic eye, in which, 
before extraction, a concave spherical lens of 20 D. 
was found to correct the ametropia. Making the 
proper substitutions, we first find that in tliis case 
the retina lies 29.7 mm. behind the cornea, and since 
the posterior focus of the aphakic eye is 31 mm. 
behind the cornea, the eye will be hyperopic. 



104 HANDBOOK OP OPTICS 

In the equation — + — = 1, I^ now represents 

the anterior focal length of the aphakic eye ; it is 
therefore approximately 23 mm. ; F^ represents the 
posterior focal length, approximately 31 mm,; /' 
represents the distance A!F (Fig. 29), and in this 
case is 29.7 mm. From these data we can find / 
(A!Q^ Fig. 29), the focal length of the required lens 
if placed in contact with the cornea. We thus de- 
rive the value/ = — 512 mm. Supposing that the 
lens will be worn in the same position as the con- 
cave lens previous to extraction, we add 15 mm. to 
512 mm. The dioptric power of the lens required 
to correct the existing hyperopia is therefore 1.9 D. 
This result agrees very closely with the condition as 
found to exist in a particular case after the removal 
of the transparent lens for the cure of myopia. So 
close an approximation could not be obtained in 
every case, however, for in high myopia there is 
frequently defect in curvature and in position of 
the lens as well as in length of the eye. 

From the foregoing studies we also learn the 
effect of a change in the position of the crystalline 
lens upon the refractive power of the eye ; if the 
lens move toward the cornea the eye will be rendered 
myopic, and if it move away from the cornea the eye 
will become hyperopic. 

We shall next study the effect of changing the 



ERRORS OF REFRACTION IQo 

position of spectacle lenses when used for near 
vision. We have seen that convex lenses used to 
aid the distant vision of hyperopes increase in power 
as they are withdrawn from the eye. When convex 
lenses are worn by presbyopes to replace the accom- 
modation which has failed, the effect of withdrawing 
the lens varies under different circumstances. Sup- 
pose first that we are dealing with an emmetropic 
eye whose accommodative power has entirely failed ; 
this eye can focus only parallel rays upon the retina. 




Fig. 31. 

Let P (Fig. 31) represent the point where a near 
object intersects the axis ; then, in order that the 
object be clearly seen by the eye without accommo- 
dation, a convex lens A^ whose focal length is AP^ 
must be placed before the eye. As the lens is re- 
moved from the eye the focal length AP diminishes, 
and the strength of the lens necessary to produce 
distinct vision must be increased. 

We shall next take the case of a hyperopic eye 
whose accommodative power has been lost. Let P 
(Fig. 32) represent the intersection of a near object 
with the axis ; then the conjugate of the point P in 



106 HANDBOOK OF OPTICS 

the refraction by the lens is Q, which lies behind the 
retina. If the position of Q be such that it is con- 
jugate to R in the refraction by the eye, then the 
object at P will be accurately focused on the retina 



with the aid of the lens. If P remain stationary 
while the lens is moved to A^\ the effect upon the 
power of the lens will not be the same in all cases. 
We have the equation 

1,1 1 1,11 

+ -7-7; = -^, or - + - = —. 



AP ' AQ F' f f F 

When the lens is moved from A to A^\ the dis- 
tance/is diminished by AA^\ and/^ is increased by 
AA' . To study the effect of this change upon the 
equation, we take the case when / and /' are equal, 
each being then equal to '2F, The equation then 

1 1^11 

becomes ^^^^^^^J 

When the lens is at A!^ we have the similar equa- 
tion 

1^1 1 



IF^AA!' 2F + AA" F^ 



ERRORS OF REFRACTION 107 

from which 

1 4F 1 



4F 

From this we have F. = F — — ^ • Therefore, 

F^ is less than F^ and the dioptric power of the lens 
must be increased in order that Q may be conjugate 
to P, In other words, for a fixed lens the line PQ 
is shorter when AF and AQ are equal than for any 
other position of the lens ; conversely, if the line PQ 
remains constant, a stronger lens is required when it 
is moved from its central position at A, whether it is 
moved toward P or toward Q. Hence we have this 
rule : If the distance between object and lens is less 
than twice the focal length of the lens, the power of 
the lens is diminished by moving it from the eye and 
toward the object ; if, on the other hand, the distance 
between object and lens is more than twice the focal 
length of the lens, the power of the latter is increased 
by moving it away from the eye. It will be observed 
that the distance at which a book is held for reading 
is ordinarily less than twice the focal length of the 
lens worn as a reading glass ; consequently the power 
of such a glass is weakened by moving it from the 
eye. 

In Fig. 33 we have a representation of an eye 
myopic either from abnormal length of axis or from 



108 



HANDBOOK OF OPTICS 



act of accommodation. Without the aid of a lens 
the conjugate of ^ is at J? on the retina ; but in 
order that an object at P, nearer than §, may be 
seen, a convex lens is required ; then in the refrac- 
tion by the lens, P and Q are conjugate. It is seen 
that AP is positive and AQ \^ negative ; therefore 
AP must be less than 2F, for it must be less than F 
in order that AQ may be negative. Hence, as in the 



Q 




previous case, the power of the lens will be dimin- 
ished by moving it away from the eye. 

From this we see that the statement that pres- 
byopes increase the power of their glasses by moving 
them to the tip of the nose is inaccurate. The power 
of the lens can be increased in this way only on con- 
dition that the object is moved in the same direction 
and to the same extent as the lens ; but as the object 
is removed from the eye, the retinal image becomes 
smaller — a manifest disadvantage. 



CHAPTER VII 

THE EFFECT OF SPHERICAL LENSES UPON THE 
SIZE OF RETINAL IMAGES 

Spherical lenses used for the purpose of bringing 
the image of an object to an accurate focus on the 
retina have also an effect upon the size of retinal 
images. In many cases this is so slight as to escape 
notice ; but when strong glasses are worn the effect 
is considerable. If the glass before one eye differs 
much in strength from that before the other eye, 
confusion results from the unequal images, and such 
glasses are usually rejected. The size of the image 
after refraction through any optical system is ob- 
tained, as we have learned, from the equation 

1 = — — ^, 

where u is the distance of the object from the an- 
terior focus of the system, and F^ is the anterior 
focal distance. This equation gives a negative value 
for the image when it is real, since such an image 
lies on the opposite side of the axis to the object, 
and is consequently inverted. As we shall have in 

109 



110 HANDBOOK OF OPTICS 

our further studies to deal only with retinal images, 
we shall neglect the minus sign, because the image 
is seen by the observer as erect, the transformation 
being made by mental act. Hence we shall consider 
as positive those images which appear to be erect ; 
and we shall consider as negative those which appear 
to be inverted. Our equation thus becomes 

u 

If now a lens be introduced before the eye, we 
shall have a new optical system ; and, as before, the 
size of the image formed by this system will be 
found from the equation 

where ^6^ represents the distance of the object from 
the new anterior focus, and F represents the new 
focal distance. Hence the relation of the size of 
the image with the lens to that without the lens 
is expressed by the equation 

^^ _ u X F 
i U^ X i^2 

When the distance of the object is great in com- 
parison with the change in position of the anterior 



EETINAL IMAGES HI 

focus caused by adding the lens, then u and u-^ may 
be considered identical. As this is true in the case 
of lenses used as spectacles, we have the simplified 
expression 






But on page 93 we learned that the anterior focal 
distance of the eye combined witli a lens is derived 
from the equation 



F 



F^ 



F^ + F,^-e . 
Making this substitution, we have 

^ = __Z__. 
i F^ + F^- e 

In this equation F^ is the focal length of the lens, 
F^ is the anterior focal length of the eye, and e is 
the distance between eye and lens. This formula 
is sufficiently accurate to determine the effect of 
spectacles upon retinal images except when the 
object is very near the eye ; but as we wish to show 
the magnifying power of lenses in all positions in 
front of the eye, we shall examine the condition 
when u and ii^ cannot be considered identical. Re- 
ferring to Fig. 28 we see that u = PF^, and u^ = FF. 



112 HANDBOOK OF OPTICS 

Hence u^ = u + FF^ = u + QAF - AF^) ; 
but on page 93 we found that AF is equal to 

F^ + F^-e 
We see also that 

From this we find 

uF^ + u(F^ - e) + (j; - ^y 

F^+F^-e 



or, 



and i^= ^ 



i Mj'i4-M(-F2-e) + (-^2-0^' 



or, 



i' _ F^ 



' F^ + iF^ - + ^^^ ~ ^^' . 



^t 



This is the general expression for the magnifying 
power of any lens in combination with the eye, or 
with any other optical system. Examining this 
expression, we see that if F^ = e^ that is, if the lens 
be placed at the anterior focus of the eye, 



RETINAL IMAGES 113 

Hence a lens so placed has no effect upon the size 
of retmal images. This is true whether the lens is 
convex or concave, and irrespective of the distance 
of the object. 

If the lens be convex and F^ be less than g, that 
is, if the lens be without the anterior focus of the 
eye, F^^ — e will be negative, but (jP^ — 6)^ will be 
positive. The least value which u can have is equal 
to ^ — F^, *since the object cannot lie between the 
lens and eye. When u = e — F^^ i' and i are equal, 
which indicates that there is no effect on the size of 
the image when object and lens are in contact. But 
as the distance of the object is increased and u 
assumes a greater value, the lens exerts a magnify- 
ing power upon the image, and this magnification 
increases as the distance between lens and object 
increases. We observe also from a further study of 
the equation that as e, the distance between eye and 
lens, varies, the magnifying power varies. When 
this distance becomes such that 



u 



the denominator of the expression which denotes the 

magnifying power becomes zero, and - becomes 
iniijiite. 

If (e - F^^ is greater than F^ + ^ '^ ~ ^^ , then ^ 
becomes negative. 



114 HANDBOOK OF OPTICS 

To render the meaning of this clear, we shall first 
suppose the object to be so distant in comparison 
with the distance between lens and eye that the 

fraction ^^— ^ ^ may be neglected as inappreciable. 

In this case — becomes — ^ , and this exyjres- 

i F^ + F^ — e 

sion is negative when e — F^i^ greater than F^^ that 
is, the image is inverted when the distance of the 
lens from the anterior focus of the eye is greater 
than the focal length of the lens. An aerial image 
of the object will then be formed by the lens in 
front of the eye, and a second image will be formed 
by the eye.* 

As the object approaches the lens, the term 

5^—2 ^ can no longer be ignored, and e — F^^ the 

u 

distance of the lens from the anterior focus of the 

/^ XT \2 

eye, must be greater than F^ + ^^ — ^ L in order 

u 

that the inverted image be formed. 

The meaning of this is that as the object ap- 
proaches the lens, its image recedes behind the pos- 
terior principal focus, F^^ and in order that the 
aerial image be formed in front of the anterior focus 
of the eye, the distance of the lens from the eye 

* When the aerial image is near the anterior focus, the final 
image will lie far behind the retina, and no distinct image will be 
perceived by the eye. 



RETINAL IMAGES 115 

must be greater than when the object is more re- 
mote. If we solve the equation 

e-F,-F, + , . 

we obtain the relation between e — F^ and u^ which 

i' . . 
exists when — is infinite. This relation thus becomes 



Since we cannot extract the square root of a nega- 

4:F 

tive quantity, the fraction 1 must be less than 1, 

or equal to it, in order to give a real value to the ex- 
pression. Hence we see that the least value which u 

can have and satisfy the condition — = oo is u = 4:F-^] 

and when u = 4:F^ we find the corresponding value 

for e — F^ to be -. In other words, the image can- 

not become infinite, and consequently cannot become 
negative, when the distance of the object from the 
anterior focus of the eye is less than four times the 
focal length of the lens. When this distance is 
equal to four times the focal length of the lens, the 
image becomes infinite provided the lens is placed 
midway between the object and anterior focus of the 
eye. All this follows from what has been said in 
the preceding chapter in regard to the effect of 



116 HANDBOOK OF OPTICS 

changing the position of a lens in near vision, and 
we need not have deduced this result algebraically. 
We have seen that PQ (Fig. 32) is shorter when the 
lens, A^ is half way between P and Q than in any 
other position. Similarly, in Fig. 28, the least dis- 
tance from F^ at which P may be situated and still 
form an image at this point is four times the focal 
length of the lens, and the lens must be at the mid- 
way point of the line PF^. Whether the lens be 
moved toward the object or toward the ej^e, the 
image will be thrown to the right of F^^ and the 
final image will be reduced in size. Thus the image 
attains its greatest size when the lens is in this mid- 
way position, and we have this rule for determining 
the effect of changing the position of the lens : As 
the convex lens is removed from the eye, the magni- 
fying power increases so long as the distance of tlie 
object from the lens is more than twice tiie focal 
length of the lens, and when the distance between 
object and lens is less than twice the focal length of 
the lens, the magnifying power is diminished by 
further removal of the lens from the eye. 

If, while the lens is convex, F^ — e be positive, 
that is, if the lens be placed within the anterior 
focus of the eye, then i^ will always be less than i. 

We next suppose the lens to be concave. In this 
case F^ is negative. Our equation shows that when 
F^ is less than g, or when the lens is without the an- 



RETINAL IMAGES 117 

terior focus, i' is less than z, except when it = e — F^; 
in tliis case, as with convex lenses, i^ and i are equal. 
When F^ is greater than e, i' is greater than i. 

To sunimarize : A convex or concave lens placed 
at the anterior focus of the ej'e or of an}' optical sys- 
tem, though it alters the j^osition of the image of an 
object, has no effect upon the size of the image. 

A convex lens placed without the anterior focus 
of an optical system magnifies the image, the degree 
of magnification varjdng with the distance between 
the lens and anterior focus, and with the distance 
between the object and the lens. 

A convex lens placed witJiin the anterior focus of 
an optical system minifies the image, the degree of 
minification varying with the distance between the 
lens and anterior focus and with the distance between 
the object and lens. 

A concave lens placed loithin the anterior focus 
magnifies the image ; if placed ivitliout the anterior 
focus it minifives the image, the degree of magnifi- 
cation or minification varying with the distance be- 
tween lens and anterior focus and with that between 
object and lens.* 



* The student should verify these phenomena by taking a con- 
vex lens of 20 dioptres, which may represent the eye ; if another 
lens be held before this and be moved to and fro while an object is 
viewed through the combination, the effect of the second lens upon 
the size of the image can be easily noted. 



118 HANDBOOK OF OPTICS 

The anterior focus of the eye we have found to 
be 13.7504 mm. in front of the cornea. Spectacle 
glasses, being usually worn farther from the eye 
than this on account of the projecting eyelashes, 
must affect the size of retinal images ; convex lenses 
magnify and concave lenses minify these images. 
Thus in the hyperopic eye, which sees with the aid 
of a convex lens, the image is larger than it would 
be in the emmetropic eye, because the optical sys- 
tem is the same in the two eyes, and the addition 
of a convex lens placed without the anterior focus 
magnifies the image. Similarly, in a myopic eye, 
which sees with the aid of a concave lens placed 
without the anterior focus, images will be smaller 
than in an emmetropic eye. 

We must now compare the image as formed by 
the unaided hyperopic eye with that formed by the 
emmetropic eye. The hyperopic eye is enabled to 
see objects by an increase of curvature of the crys- 
talline lens, by which means its dioptric power is 
increased and its focal distances are diminished. 
Hence the size of the retinal image is diminished 
by the change, since this is proportional to the an- 
terior focal distance. The hyperopic eye without 
a correcting lens has smaller images than the emme- 
tropic eye ; and as the same eye with a lens has 
larger images than the emmetropic eye, it is evident 
that in hyperopia images are increased in size by the 



RETINAL IMAGES 119 

correcting lens. The same holds true in near vision, 
since a greater amount of increase in curvature is 
necessary for the hyperopic eye to see near objects 
than for the emmetropic eye to see the same objects, 
and in consequence the image in the hyperopic eye 
is smaller than in the normal eye ; but with the aid 
of the correcting lens the image is larger in the 
hyperopic than in the normal eye. Likewise, pres- 
byopes, who wear convex lenses to enable them to 
see near objects, have larger images than those who 
see near objects by act of accommodation. 

The myopic eye cannot adapt itself to distant 
vision, and hence without a lens all images of dis- 
tant objects are blurred. Such an eye, however, 
sees near objects either without any increase of 
curvature or with less increase than would be re- 
quired in an emmetropic eye. Thus we see that 
the myopic eye has the advantage of larger images 
of near objects than the normal eye has. If the 
myopia is corrected by a concave lens, the eye must 
now make the same effort of accommodation that the 
normal eye makes. The image is accordingly mini- 
fied by the increase of curvature and by the concave 
lens if this is worn without the anterior focus of the 
eye. In the high degrees of myopia, in which strong 
glasses are required, the minifying effect is a serious 
obstacle to their use. In these cases the lenses 
should be worn as near the eyes as possible. 



120 HANDBOOK OF OPTICS 

We shall now investigate the aphakic eye. The 
second principal focal distance of this eye is 
31.095 mm. As the retina is only 22.8326 mm. 
behind the cornea, we have in the aphakic eye a high 
degree of curvature hyperopia. The anterior focal 
distance of the aphakic eye is 23.2659 mm. Hence 
the eye which has been deprived of its crystal- 
line lens has larger images than the normal eye. 
The image, however, will be formed far behind the 
retina ; and in order to bring it forward to the 
retina a strong convex lens is required. This lens 
worn as a spectacle glass will be within the anterior 
focus of the aphakic eye. It will, accordingly, 
reduce the size of images so that they more nearly 
correspond in size with those as formed in the nor- 
mal eye. If the eye be hyperopic prior to the 
extraction of the lens, the images will be still more 
nearly approximated in size to those formed in 
the normal eye, since a stronger convex lens will 
be required to focus rays on the retina than if the 
eye had been normal. But even in high degrees 
of hyperopia the image will not be reduced by the 
correcting lens to the size of the normal image. 

In the myopic eye images w^ill be larger than if 
the eye had been emmetropic before extraction, 
because a less convergent lens will be required than 
if the eye had been normal. Thus we see that in 
all aphakic eyes the retinal images, as formed with 



RETINAL IMAGES 121 

the aid of correcting lenses, are larger than in nor- 
mal eyes, and that of aphakic eyes, those which 
were hyperopic prior to the extraction are least 
favorably sitnated as regards the size of images, 
and those which were myopic prior to the extraction 
are most favorably situated in this respect. To this 
enlargement of images is due the remarkable increase 
of visual acuity after extraction of the lens for the 
cure or improvement of myopia. We have seen 
that the excessively myopic eye which requires a 
strong concave lens to produce clear images is very 
much hampered by the minifying effect of the cor- 
recting lens. If, after the extraction of the crj^stal- 
line lens, parallel rays of light are focused on the 
retina without the aid of a lens, the size of the image 
of a distant object, as formed in this eye, will be 
to the size of the corresponding image m a nor- 
mal eye as the anterior focal distance of the aphakic 
eye is to that of the normal eye. Omitting fractions 
this ratio is as 23 to 15.* When we consider that, 
before the extraction, images were reduced in size, 
we are prepared to expect great improvement in 
visual acuity in those cases in which the operation 
has been successfully accomplished. 

To the enlargement of images, though in a less 

* This ratio refers to the linear dimensions of the images. The 
relative amount of retinal surface covered by the images is pro- 
portional to the squares of these numbers. 



122 HANDBOOK OF OPTICS 

degree, is also due the excellent visual acuity which 
sometimes follows cataract extraction in emmetropic 
eyes. In the most successful cases a visual acuity 
of ^^ may be obtained in spite of the fact that the 
pupil is not entirely free from particles of opaque 
matter. 

Finally, we shall consider the effect upon retinal 
images of changing the position of the convex lens 
in near vision. We have seen, as we might infer, 
that the size of the image varies according to the 
same rule as does the dioptric power of the lens 
in any position, that is, if the distance between the 
object and lens is less than twice the focal length 
of the lens, the power of the lens is weakened 
and the size of the image is diminished by removal 
of the lens from the eye ; and if the distance of 
the object is more than twice the focal length of 
the lens, the power of the lens and the size of the 
image are increased by removal of the lens from 
the eye. Thus, as the prevalent belief that pres- 
byopes increase the power of their glasses by placing 
them on the tip of the nose was shown to be not 
generally true, so the statement made in some text- 
books that by so placing them larger images are 
obtained, is also erroneous. 

The condition when the eye is emmetropic and 
without accommodative power is shown in Fig. 31. 

When one looks through an ordinary hand magni- 



KETINAL IMAGES 123 

fying glass, tlie same conditions are present. If, as 
is usually the case, the distance of the object from 
the anterior focus of the eye is less than four times 
the focal length of the lens, the image is never in- 
verted. If this distance is less than twice the focal 
length of the lens, then as the lens is removed from 
the eye the image is diminished in size. But when 
the lens A is farther from the object P than the 
focal length of the lens, tlie pencils of light after 
passing through the lens are convergent, and cannot 
be focused by a normal eye ; hence the nearest point 
to the eye at which the lens can be placed and afford 
distinct vision is such that the object is at the focus 
of the lens, and this is for the emmetropic eye the 
position of greatest magnifying power. The hyper- 
opic eye can focus the convergent pencils and receive 
a clear image when the lens is to the right of A ; on 
the other hand, the myopic eye can only focus diver- 
gent pencils such as are formed when the lens is on 
the left of A and nearer to the object. 



CHAPTER VIII 

CYLINDRICAL LENSES 

We have assumed that the refracting surfaces of 
the eye are spherical in form. This is permissible 
in normal cases, but in a large proportion of eyes the 
refracting surfaces — more especially the cornea — 
do not have this regularity of curvature ; the curva- 
ture is found to vary appreciably in different meridi- 
ans, the meridians of greatest and least curvature 
being usually at right angles to each other. These 
are called the principal meridians. Such a surface 
is called a torus or toric surface. The outer convex 
surface of a ring is an example of a toric surface. 
In an eye whose cornea is of this form the image of 
a point will not be another point, and from this fact 
the defect is called astigmatism. This asymmetry of 
refraction was first noticed by Dr. Thomas Young, 
the celebrated physicist, but Sir George Airy, who 
had the defect in his own eyes, was, in 1827, tlie 
lirst to correct it by means of suitable lenses. 

Astigmatism may be produced by faulty curvature 
or oblique position of the crystalline lens, but its 
most common source is asymmetrical curvature of 

124 



CYLINDRICAL LENSES 125 

the cornea. If the curvature of the cornea be nor- 
mal in one meridian and too little or too great in the 
meridian at right angles to this one, the defect may 
evidently be corrected by a lens having no curvature 
in the normal meridian, and having in the meridian 
at right angles to this such curvature as will counter- 
act the defective curvature of the cornea in this me- 
ridian. A lens of tliis nature would be cylindrical 
in form. If, with the equalization of curvature, the 
eye is still ametropic, a sj^herical lens may be com- 
bined with tlie cylindrical lens. The spherical 
surface may be ground on one side of the glass 
and tlie cylindrical on the otlier ; or a toric curva- 
ture may be ground on one side of the glass, leav- 
ing the other side plane ; and, if desired, this may 
then be made concave and the lens periscopic. 
Toric lenses are not much used, as they are more 
difficult to grind than spherical and cylindrical 
surfaces. 

In a cylindrical lens the line drawn through the 
summit of curvature and parallel to the axis of the 
cylinder is called the axis of the lens.* Reference 
to Fig. 3-i renders it clear that in the direction at 
right angles to the axis AA^ of the lens a cylindrical 
lens has the same action tliat a spherical lens of like 
radius of curvature and index would have in this 

*This must not be confounded with the axis of the optical 
system. 



126 



HANDBOOK OF OPTICS 



direction, and that the deviating power of the lens 
is confined entirely to this meridian. Rays of light 
parallel to the axis 00' passing through such a lens 
would not be united in a focus. They would, how- 
ever, all meet a line which is parallel to the axis of 
the lens, and whose distance from the lens is equal 
to the focal length of a spherical lens of the same 




radius of curvature and index. This distance is 
called the focal distance of the cylindrical lens, and 
the line is called the focal line. Similarly, rays 
diverging from a point will, after refraction by the 
lens, all pass through a line conjugate to the point 
from which the rays proceed. 

If we take another cylindrical lens whose axis is at 
right angles to that of the first lens, it will bring all 



CYLINDRICAL LENSES 



127 



the rays from a point into a line at right angles to 
the first line.* 

In Fig. 35 the rays from the point P all pass 
through the line AB ; and rays from P^ all pass 
through the line A' B' at right angles to AB, 
Hence, if these two lenses have equal refractive 




Fig. 35. 



power, and if we combine them so that the rays after 
deviation by the first lens pass immediately through 
the second lens, the effect of both lenses will be to 
cause the rays to meet in the point where the lines 
intersect ; for the rays must all pass through both 
lines. In other words, two equal cylindrical lenses 
placed with their axes at right angles are equivalent 



* It will be understood that we are bound by the same restric- 
tions as in the case of spherical lenses, that is, we must suppose 
that only rays near the axis of the pencil pass through the lens. 



128 



HANDBOOK OF OPTICS 



to a spherical lens of the same index and radius of 
curvature. In Fig. 36 let P represent the point at 
which a ray parallel to the axis of the refracting 
system meets a spherical lens ; then when this ray 
reaches the focus of the lens it will have been so 
deviated as to pass through the axis at F. The mo- 
tion of a point from P to F may be considered as the 



.iN 






M. 







i \ 



^F 



Fig. 36. "^^^-^v' 

resultant of three motions at right angles to each 
other: 1st, from P to M; 2d, from M to 0; and 
3d, from to F. In the case of the spherical lens 
the deviation in the plane PMOJY wiW be represented 
by the line PO; if instead of a spherical lens we 
take two equal cylindrical lenses with axes at right 
angles, then in this plane the ray will be deviated by 
one lens from P to Jf, and by the other from Mto 0. 



CYLINDRICAL LEMSES 



129 



The result is the same 
in the two cases. If the 
two cylindrical lenses have 
different focal lengths, 
their combined effect will 
not be equivalent to that 
of a spherical lens, be- 
cause their focal lines 
will not intersect. If 
the focal length of the 
first lens is less than that 
of the second, it is clear 
that when a ray parallel to 
the axis of the refracting 
system has been brought 
by the first lens to its in- 
tersection with the first 
focal line, it will not yet 
have reached its intersec- 
tion with the second focal 
line. 

Let C (Fig. 37) repre- 
sent a combination of two 
cylindrical lenses whose 
axes are vertical and hori- 
zontal respectively. The 
line AB is conjugate to P 
as regards the lens whose 




130 HANDBOOK OE OPTICS 

axis is vertical ; and A^B\ at right angles to AB^ 
is conjugate to P as regards the lens whose axis 
is horizontal. We take two rays, PR and PR^^ 
so situated that they pass through the axis of the 
second lens. Hence they will be deviated only 
by the first lens ; they will intersect at E in the 
first conjugate focal line, and after crossing at this 
point they will meet the second conjugate focal line 
at 2^ and T' , Likewise, if we take two rays, PS and 
PS ^ which pass through the axis of the first lens, 
they will be deviated only by the second lens and will 
intersect at W in the second conjugate focal line, 
meeting the first conjugate focal line before intersec- 
tion at /and I' . All rays which do not pass through 
the axis of either lens will be deviated by both lenses, 
and the deviation by one lens will be superposed upon 
that of the other. The action of the first lens is such 
as to cause all rays from P to pass through the verti- 
cal line AB conjugate to P ; and since the second 
lens deviates light in the vertical meridian, its effect 
is to change the position of rays in the line AB^ but 
not to deviate them out of this line ; hence all rays 
from P must pass through AB, Similarly, the action 
of the second lens is such as to cause all rays from P 
to pass through the horizontal line A^ B^ , and since 
the first lens deviates light only in the horizontal 
meridian, its effect is to change the position of rays 
in the line A^ B\ but not to deviate them out of this 



CYLINDRICAL LENSES 131 

line. From this we see that rays proceeding from a 
point will never be nnited in a focus by such a lens ; 
they will, however, all pass through two lines which 
are at right angles to each other, called focal lines. 
If the point is so far distant that the rays may be 
regarded as parallel, the lines AB and A^B^ are the 
principal focal lines. The interval between the prin- 
cipal focal lines is called the focal interval of Sturm.* 

It is clear that the more nearly equal the two cylin- 
drical lenses are in power, the shorter will be the 
focal interval, and the more closely will the image of 
a point resemble a point. 

Since two equal cylindrical lenses with axes at 
right angles are equivalent to a spherical lens, it 
follows that two unequal cylindrical lenses, similarly 
combined, are equivalent to a sphero-cylindrical lens ; 
for we may regard the unequal lenses as composed 
of two equal cylindrical lenses with the addition of 
another cylindrical lens. 

In order to investigate the image of a point after 
refraction by an astigmatic surface, we must suppose 
a screen to be placed at different positions in the 
path of the rays so as to intercept them. If a 
screen be placed at AB^ the image of the point P, 
as formed on the screen, will be the vertical line JP. 

*The theory of refraction by asymmetrical surfaces was first 
demonstrated by Sturm in 1845. See " Comptes Rendus de I'Acad. 
des Sci. de Paris/' torn, xx., pp. 554, 761, 1238. 



132 HANDBOOK OF OPTICS 

Likewise, the image of the point as formed at A^B^ 
is Tr. To the left of AB the image will be ellip- 
tical, for the rays will not yet have intersected 
either focal line, but they will be nearer the vertical 
than the horizontal intersection. To the right of 
A'B^ the image will be elliptical, for the rays have 
passed both vertical and horizontal intersections, but 
are nearer to the horizontal than to the vertical 
intersection. Between AB and A^B^ the image will 
also be elliptical, except in one position, in which 
the distance of the rays from the axis is the same 
in the vertical as in the horizontal meridian ; it will 
then be a circle. This is called the circle of least 
confusion. To the left of this circle the ellipse has, 
in our case, the long axis vertical, and to the right 
the long axis is horizontal. 

The image of a line will vary according to its 
position in relation to the focal lines. If the line 
be a vertical one passing through P, then at AB 
every point of the line will have as its image a 
vertical line such as IF. The image of the vertical 
line will be a lengthened and intensified line. At 
A' B\ however, every point of the vertical line will 
have as its image a horizontal line such as TT' ^ and 
the image of the vertical line will be an aggregation 
of horizontal lines ; it will therefore be a hroad and 
indistinct line. At any other point the image will con- 
sist of a superposition of confusion ellipses or circles. 



CYLINDRICAL LENSES 133 

If the line be horizontal, we shall have a broad 
blurred image at AB and a clear intensified image 
at A'B\ If the line be neither vertical nor hori- 
zontal, its image will be blurred in all positions. 

From what has been said we see that a vertical 
line appears distinct at the focus of the cylindrical 
lens whose axis is vertical, and a horizontal line ap- 
pears distinct at the focus of the lens whose axis is 
horizontal; or, since the meridians of refraction are 
at right angles to the axes, a vertical line appears 
distinct at the focus of the horizontal meridian, and 
a horizontal line appears distinct at the focus of the 
vertical meridian.* 

The formulae which we have deduced for spherical 
lenses are also applicable to cylindrical lenses. We 
have only to bear in mind that the action of the 
latter is confined entirely to the meridian at right 
angles to the axis of the lens. What has been said 
of the effect of spherical lenses upon the size of reti- 
nal images applies, therefore, to cylindrical lenses, 
with the understanding that this effect is confined 
to the refracting meridian, no eft'ect being produced 
by the cylindrical lens in the meridian of its axis. 

Astigmatism of the eye is a curvature defect, 

* The student who finds it difficult to comprehend refraction 
through the double cylindrical or sphero-cylindrical lens should 
construct for himself, or procure from an instrument maker, thread 
models, which illustrate very clearly astigmatic refraction. 



134 HANDBOOK OF OPTICS 

while hyperopia and myopia are, in a large majority 
of cases, axial defects. The effect of hyperopic or 
myopic astigmatism upon retinal images will, con- 
sequently, not be analogous to that of axial hyper- 
opia or myopia. If the eye is hyperopic in one 
meridian and emmetropic in the meridian at right 
angles to this, the defect in curvature in the hyper- 
opic meridian is the same as if a concave cylindrical 
lens were placed in contact with a normal cornea. 
The effect of such a lens, since it would be within 
the anterior focus of the eye, would be to enlarge 
images in the refracting meridian of the lens. In 
other words, since the curvature of the eye is less 
in the hyperopic than in the emmetropic meridian, 
the anterior focal distance is greater in the faulty 
than in the normal meridian ; and consequently the 
image of an object will be too large in the former 
meridian, for the size of the image is proportional 
to the anterior focal distance. Similarly, in myopic 
astigmatism, the image is too small in the myopic 
meridian. But we must remember that the retina 
is not in the proper position to receive an accurately 
focused image in the faulty meridian ; and, to dem- 
onstrate the effect of this faulty position of the 
retina, let (9, Fig. 38, be the optical centre of a lens 
which may, for the present purpose, represent the 
eye ; then, if A be conjugate to P, the image oi PQ 
will be AB, If, now, the screen or retina remain at 



CYLINDRICAL LEXSES 



135 



A, while the lens is increased in po\Yer so that P^ is 
conjugate to P, the true image, P' Q' ^ will be smaller 
than AB, but the indistinct image, as intercepted by 
the screen, will be AC, which is larger than AB,^ 

This is why a round object such as the fall moon 
appears greater in the meridian of myopic refraction 
than in the emmetropic meridian, though the accurate 
image is less in the myopic meridian. In like man- 



Q. 


^^^^ 






c 




^^^^r^^^ 


«> 


^ 


B 


_p 


^^^%< 


^\y 




A K 




u^^^ 




^^ 


^^^^^ 1 


Q 


Fig. 38. 






5^, 



ner, if the screen, remain at A while the conjugate of 
P is changed from A to P^, it is apparent that by 
the change both the focused and blurred images will 
be rendered larger than the normal, but the blurred 
image will be less enlarged than the other. 

We can now understand the influence upon retinal 
images of cylindrical lenses used as spectacles. A 
proper convex cylindrical lens worn as a spectacle 



* The change in position of the optical centre, being too slight 



to affect the resalt, is neglected in the figure. 



136 HANDBOOK OF OPTICS 

brings the image of an object to an accurate focus on 
the retina ; but it also enlarges the image in the 
meridian at right angles to the axis of the lens. For, 
as we have seen in Fig. 38, if the lens be worn at the 
anterior focus of the eye, the new image will be of 
the same size as P^ Q^^ since the effect of the lens is 
to bring the image forward without changing its size. 
If the lens be worn without the anterior focus of the 
eye, then the new image will be larger than Pi Qi ; 
in either case it will be larger than the blurred image 
AO'^ which the eye receives without the lens, and 
larger than AB' ^ the normal image. A proper con- 
cave cylindrical lens throws the image back upon the 
retina ; it also minifies the image in the refracting 
meridian of the lens. If the lens be worn at the an- 
terior focus of the eye, P^ Q' will represent the size of 
the new image ; and if the lens be worn without this 
focus, the image will be smaller than P^Q' ; in either 
case it will be smaller than the blurred image AO^ 
which the eye receives without the lens, and smaller 
than AB^ the normal image. Hence we see that 
cylindrical lenses worn as spectacles do not, under 
any circumstances, produce normal retinal images ; 
all objects are magnified in the refracting meridian 
by a convex lens, and minified in this meridian by a 
concave lens. If the lens could be worn in contact 
with the cornea, the seat of defective curvature, nor- 
mal images would result. The same position of the 



CYLINDRICAL LENSES 137 

spherical lens would be necessary to produce images 
of normal size in curvature hyperopia and myo]3ia ; 
but, as shown in the preceding chapter, in ordinary 
axial hyperopia and myopia, the lens must be placed 
at the anterior focus of the eye in order that images 
of normal size be produced. 



CHAPTER IX 

THE TWISTING PROPERTY OF CYLINDRICAL LENSES 

As a consequence of the effect of cylindrical lenses 
upon retinal images, it follows that if one hold such 
a lens in front of the eye, and through the lens look 
at a distant rectangular object as a picture frame or 
test-type card, there will be observed a distortion of 
the object, which will vary with every variation in 
the position of the lens. If the axis of the lens be 
parallel to one of the sides of the object, the rectan- 
gular form of tlie object will be retained, but the 
ratio of the sides will be altered. The side which is 
parallel to the axis will not be changed, wdiile tliat 
which is perpendicular to the axis will be increased 
or diminished. If now the lens be rotated in its own 
plane, the distortion will no longer be confined to 
the apparent size of the object ; it will also affect the 
direction of the lines forming the sides, so that the 
rectangular object will assume the form of an oblique 
parallelogram. Use is made of this phenomenon to 
determine the position of the axis of a cylindrical 
lens. Looking through the lens at a distant straight 
line, we observe the position of the lens in which 

138 



CYLINDRICAL LENSES 



139 



there is no apparent deviation of tlie line ; the axis 
of the lens must be either parallel or perpendicular to 
this line. 

We have seen that a cylindrical lens has in its 
refracting meridian the same effect that a similar 
spherical lens would have. The formula by which 
the magnifying or minifying power is obtained has 
been given in Chapter VII. (page 112). 




Fig. 39. 



Let OA (Fig. 39) be a line parallel to the axis of 
a cylindrical lens, and let BA be perpendicular to 
the axis. If we look through the lens at the line 
OA^ the image of this line on the retina will be the 
same as without the lens. If we look at the line 
AB^ and if the lens be convex, then AB will be 
magnified in its own direction, and will appear as 
the line AB^. If we now look at an oblique line 
OB, its direction will be changed, for B will apjDcar 
at B' and ^ will appear at ^^ and so on for every 



140 HANDBOOK OF OPTICS 

other point of the Ime. The line OB will conse- 
quently assume the position OB' , Hence any line 
not parallel or perpendicular to the axis of the lens 
undergoes an angular deviation when viewed through 
a cylindrical lens. If a is the angle which the line 
makes with the axis, and x the angle which the line 
appears to make with the axis, we have 

AB ^ AB^ 

tan a = — -- ; tan x = -—— ; 

OA OA 

AB' 

from which tana: = ———tana. 

AB 

AB' 

being the magnifying power of the lens may 

AB 

be called m. Then tan x = m tan a. When the lens 

is concave m is less than unity, and the line OB' 
appears as OB, 

Taking a convex lens, we look through it at a 
pencil held at arm's length from the eye. When 
the pencil is parallel to the axis of the lens there is 
no apparent deviation, but as we turn the pencil 
througli an angle a it appears to make the greater 
angle x with the axis of the lens. At first x in- 
creases more rapidly than a, and x — a^ which is tlie 
angle made by the apparent position with the real 
position, becomes rapidly greater. As the pencil is 
turned farther, the angle x — a changes more slowly, 
then comes to a standstill ; and finally a increases 



CYLINDRICAL LENSES 141 

more rapidly than x^ and a; — a diminislies, so that when 
the pencil has been turned through 90 degrees a has 
overtaken x and there is no deviation. If for the 
pencil we substitute a distant straight line and turn 
the lens while the line remains stationary, Ave shall 
have the same effect. As we turn the lens to the 
right, the line appears deviated to the left, since tlie 
apparent position always makes a greater angle with 
the axis than the real line. After reaching a maxi- 
mum deviation the apparent line now travels back- 
ward, coinciding with the real position when the 
lens has been turned through 90 degrees. If we 
take a concave lens, m being less than unity, the 
angle which the apparent position of the line makes 
with the axis will be less than that which the real 
position makes ; consequently, when we look at a 
distant line and turn the lens to the right, the ap- 
parent position also moves to the right, and after 
reaching a maximum deviation it travels backward, 
coinciding with the real position when the lens has 
been turned through 90 degrees. 

This to-and-fro deviation, familiar to every one who 
uses the oculists' trial lenses, vanishes under cer- 
tain conditions when we experiment with the convex 
lens. If we take a convex cylindrical lens of four 
dioptres and, holding it about 1 metre in front of 
the eye, look at a vertical line across the room, we 
shall have, upon turning the lens, the to-and-fro 



142 HANDBOOK OF OPTICS 

motion of the line ; but as we move the lens farther 
from the eye, the line becomes so indistinct that 
we cannot determine its movement ; continuing to 
increase the distance between the eye and lens, we 
now notice that as we turn the lens the behavior 
of the observed line has entirely changed. It no 
longer moves to and fro, but as the axis of the lens 
is turned through 90 degrees the apparent position 
of the line moves through twice this angle, or 180 
degrees. This phenomenon can be most easily ob- 
served by holding a pencil or similar object at arm's 
length and viewing it through a cylindrical lens of 
ten or twelve dioptres held about ^ metre in front 
of the eye. 

To explain this we shall compare the action of the 
cylindrical lens with that of the spherical lens of the 
same radius of curvature and refractive index. It 
has been shown that if a convex spherical lens be 
held before the eye at a sufficient distance, an in- 
verted image of an object Avill be formed in the air in 
front of the eye, and a second image will be formed 
by the eye. 

In Fig. 40 let (1) represent a rectangular object 
such as a picture frame ; then if it be viewed through 
a spherical lens held beyond its focal length from the 
eye, (2) will represent the object as it will appear 
to the observer. If we replace the spherical lens 
by a similar cylindrical lens with axis vertical, (3) 



CYLINDRICAL LENSES 



143 



will represent the object as it will appear to the ob- 
server. The cylindrical lens has the same effect as 
the spherical one in deviating the ra3"s in the merid- 
ian at right angles to the axis of the lens, that is, 
rays from the right of the object are made to cross 
over and intersect on the left, and vice vei^sa. The 
object therefore appears reversed in this direction ; 
but it is not reversed in the meridian parallel to the 
axis of the lens ; in this meridian the rays are un- 
affected by the lens. Similarly, (-i) represents the 



(1) 



(3) 



(3) 



(4) 



Fig. 40. 



object as it would appear when viewed through a 
cylindrical lens with axis horizontal. 

Tlie image as formed with the cylindrical lens 
will be much blurred, because the rays will, in one 
meridian, enter the eye diverging from the aerial 
image ; in the meridian at right angles to this 
they have been unaffected by the lens, and con- 
sequently diverge from the more distant object. 
They cannot, therefore, be accurately focused on 
the retina. 

To this reversal of the object in one meridian 
is due the apparent rotation of a line through 



144 



HANDBOOK OF OPTICS 



180 degrees, while the lens is turned through 90 
degrees.* 

In Fig. 41 let OB represent a line making the 
angle a with the line OA, which is parallel to the 
axis of the lens. From what we have just shown, it 




Fig. 41. 



is clear that if we view the figure AOB through a 
convex cylindrical lens so placed as to form the aerial 
image, then AB will appear in the position AB\ and 
OB will appear at OB^ , Hence, if we suppose the 
original position of the axis to be parallel to OB^ in 
this position there will be no apparent deviation of 



* Dr. Carl Roller, the discoverer of the anaesthetic property of 
cocaine, has demonstrated this phenomenon by means of analytical 
geometry. See Graefe's "Archives," 1886. 



CYLINDRICAL LENSES 145 

the line OB^ but when the axis of the lens is turned 
into the direction OA. the apparent position of OB is 
changed from OB to 0B\ The angle AOB\ which 
the apparent position of the line makes with the axis 
of the lens, is as before x. From the figure it is seen 

that tan x = — -— tan a ; where -— — represents the 
AB AB 

magnifying power of the lens at right angles to its 
axis. Hence, tan x = m tan a. This is the same 
equation as that derived for the to-and-fro deviation, 
but the angle which the apparent position OB' makes 
with the real position OB is now equal to the sum of 
X and a, whereas in the former case it was equal to 
their difference. Since ??^ becomes negative when the 
reversed aerial image is formed, x is also negative, 
and adding a negative value is equivalent to sub- 
tracting a positive value. When a is zero, x is also 
zero ; as a increases, the negative value of x increases ; 
and when a is equal to 90 degrees, x is equal to — 90 
degrees. That is, if we turn the axis through 90 de- 
grees from OB to OA, in so doing the line OB will 
be apparently rotated through 180 degrees and will 
appear in the position OB, 

It was sliown in Chapter VII. that the reversed 
image of a distant object is formed when the distance 
of the lens from the anterior focus of the eye exceeds 
the focal distance of the lens, but that as the object 
approaches the lens the distance between the eye and 



146 HANDBOOK OF OPTICS 

lens must be increased in order that the negative 
image be formed. Hence, if a pencil or similar 
object be held so that we have the continuous devia- 
tion through 180 degrees, and if the pencil be then 
moved nearer the lens, a point will be reached at 
which this phenomenon will disappear, and upon 
further approximation of the pencil to the lens, the 
to-and-fro deviation will be seen. Since spectacle 
lenses are never worn beyond their focal distance 
from the eye, it is clear that, so far as this continuous 
deviation is concerned, nothing analogous occurs in 
the use of cylindrical spectacles. 

As was shown in the preceding chapter, retinal 
images in astigmatic eyes are not normal in their 
proportions ; Ave have seen that the effect of astig- 
matism upon images is similar to that of a cylindri- 
cal lens placed in contact with a normal cornea. 
A cylindrical lens so placed would have a magnify- 
ing or minifying effect on images in the refracting 
meridian of the lens ; and, since upon this property 
depends the apparent deviation of lines, it is clear 
that in astigmatic eyes all lines not parallel or per- 
pendicular to the axis of the astigmatism are twisted 
out of their proper relations. A rectangle whose 
sides do not correspond in direction with the merid- 
ians of greatest and least refraction, appears as an 
oblique parallelogram. This distortion is, however, 
slight, for the dioptric power of the eye is great in 



CYLINDRICAL LENSES 147 

comparison with the amount of astigmatism. The 
defect is not appreciable to the person whose eyes 
are astigmatic, even if the astigmatism is of high 
degree ; but when the astigmatism is corrected by 
a suitable lens, complaint is frequently made of 
annoying distortion of lines. This annoyance is 
fortunately transitory. Since a cylindrical lens, as 
worn before the eye, cannot reduce the retinal image 
to its proper proportions, it is evident that it cannot 
correct the distortion of lines ; furthermore, by 
recalling what was said on this subject in the pre- 
ceding chapter, it will be seen that the effect of the 
lens is to increase the distortion in hyperopic astig- 
matism ; in myopic astigmatism the distortion pro- 
duced by the lens is in the opposite direction to that 
existing in the blurred image without the lens. In 
either case it is easy to see why annoyance should 
arise when glasses are first worn.* 

* Aside from this actual distortion of tlie retinal image, there 
frequently occurs a distortion due to mental influence. Eor the 
explanation of this peculiar phenomenon the following articles may 
be consulted : 

"The Effect of a Cyl. Lens with Yert. Axis Placed before One 
Eye," by 0. E. Wadsworth, Trans. American Oplith. Soc, 1875; 
''Binoc. Metamorphopsia," Archives of Ophthalmology^ 1889, and 
"New Tests for Binoc. Vision," Trans. American Ophth. Soc, 
1890, by J. A. Lippincott ; " Stereo. Illusions Evoked by Prismatic 
and Cyl. Spec. Glasses," by John Green, Trans. American Ophth. 
Soc, 1889. 



CHAPTER X 

THE SPHERO-CYLINDRICAL EQUIVALENCE OF 
BI-CYLINDRICAL LENSES 

It was shown in Chapter VIII. that two cylin- 
drical lenses whose axes are at right angles are 
equivalent to a spherical lens if the two lenses have 
the same dioptric power, and to a sphero-cylindrical 
lens if the two lenses differ in power. It remains 
now to find the equivalent of any two cylindrical 
lenses whose axes are not at right angles. Formerly 
it was not uncommon for oculists to prescribe 
glasses consisting of a combination of two cylindri- 
cal lenses obliquely inclined to each other. This 
was done notwithstanding that attention had already 
been called to the fact that such a combination is 
equivalent to a sphero-cylindrical lens. Sir G. G. 
Stokes, a celebrated English physicist, first demon- 
strated this problem, at least for the special case 
in which two lenses of equal but opposite curvature 
are combined. He devised this combination as a 
test for astigmatism. The two lenses are placed 
the one over the other so that by rotating one lens 
any angle between the axes can be obtained. This 

148 



BI-CYLI]ST)RICAL LENSES 149 

combination is called the Stokes lens. To use it 
as a test for astigmatism it should be placed before 
the eye to be examined, and the angle bet\yeen the 
axes varied until the position of best vision is found, 
when further improvement can usually be obtained 
by adding a spherical lens. From a table con- 
structed by calculation the amount of astigmatism 
can be deduced.* 

Bonders in his book on refraction refers to Stokes' 
demonstration, and also presents a solution appli- 
cable for any two cylindrical lenses ; but while his 
conclusion is correct, the demonstration is defective ; 
for, as will be subsequently shown, his assumption is 
not generall}^ true. In 1886 Dr. Jackson, of Phila- 
delphia, read before the American Ophthalmological 
Society a complete demonstration of the problem of 
cylindrical refraction, and at the same meeting Dr. 
Gustavus Hay, of Boston, offered a somewhat dif- 
ferent solution.! In 1888 Prentice J published a 
solution, similar in principle to those of Jackson and 
Hay, and in 1893 Dr. Weiland, § of Philadelphia, 
published a solution based upon that of Donders, but 

* The " Stokes lens" is chiefly of historical interest ; it is rarely 
if ever used as a test for astigmatism at the present day, for other 
more convenient tests have supplanted it. 

t Trans. American Oplitli. Soc, 1886. 

t Dioptric Fnrmiilce^ Cylindrical Lenses, 1888. 

§ Archives of Ophthalmology, Vol. XXII., No. 4, and Vol. 
XXIII., No. 1. 



150 HANDBOOK OF OPTICS 

this, containing the same error as that of Bonders, 
is not a general solution. There is also given in 
Heath's '•' Geometrical Optics " * a solution by means 
of analytical geometry, which leaves nothing to be 
desired except that this method is unsuitable for the 
use of students who are not familiar with the higher 
mathematics. Thus we see that this question has 
not lacked investigation. With our more accurate 
methods of examination it is of less practical im- 
portance than formerly, for it is seldom necessary in 
testing the vision of an eye to resort to two cylin- 
drical lenses obliquely inclined ; moreover, if, in any 
case, it should be found convenient to place two 
lenses in this manner, the sphero-cylindrical equiva- 
lent can be found without formulae or calculation. 
Nevertheless this subject must always be of interest 
to the oculist who wishes to have a scientific knowl- 
edge of optics, and therefore, before giving the prac- 
tical method of determining the equivalent in any 
case, we shall demonstrate that any two cylindrical 
lenses acting in combination are equivalent to two 
other cylindrical lenses whose axes are at right 
angles to each other, and consequently to a sphero- 
cylindrical lens. Our apology for adding still 
another solution to the list of those already pub- 
lished is that in all these solutions the process of 

* ^'Geometrical Optics," Heath, 2d ed., p. 186. 



BI-CYLIXDRICAL LENSES 151 

eliminating the special point at which the ray meets 
the lens is unnecessarily tedious. 

In studying refraction by two cylindrical lenses at 
right angles to each other, we learned that a ray of 
light in its progress from a point, P, on a spherical 
lens, Fig. 36, to its intersection with the axis at the 
focus, F, undergoes the same change in position as 
if it Avere first moved from P to in the plane of 
the lens, and then moved along the axis from to F, 
Furthermore Ave learned that the change of position 
from P to 0, in the plane of the lens, is identical 
with the resultant of the two motions from P to M 
and from M to 0, which would be produced by two 
equal cylindrical lenses at right angles to each other. 
In the same w^ay w^e may study the result of the 
deviation of a ray of light b}^ two cylindrical lenses 
obliquely inclined. In Fig. 42 let AO and BO rep- 
resent the axes of the two lenses; then a ray per- 
pendicular to the plane of the lens, and meeting the 
lens at P, would, if acted upon by the first lens only, 
be so deviated in the meridian, Pi¥, as to intersect 
the principal focal line of the lens at the distance 
F from the lens. The motion of the ray from A to 
F (P is not represented in the figure) is equivalent 
to the resultant of the two motions from P to M 
and from M to F. Similarly this ray, if acted upon 
by the second lens only, would be deviated in the 
meridian PiV, and would intersect the focal line of 



152 HANDBOOK OF OPTICS 

this lens at a distance F^ from the lens. Now each 
of these displacements, PM and PN^ in the plane of 
the combined lenses, is equivalent to two displace- 
ments at right angles ; * thus the displacement from 
P to il!f is the resultant of the displacements from 



Fig. 42. 

P to E and from E to M^ and the displacement 
from P to iV^ is the resultant of the displacements 
from P to JTand from Hio N. If the two lenses act 
in combination, the deviation produced by one lens 
must be superposed upon that produced by the 
other. Each lens produces a certain deviation in 

* Since we neglect the thickness of the lenses, they both lie in 
the same plane. 



BI-CYLINDRICAL LENSES I53 

the direction PK or LO, and also a deviation in tlie 
direction PK' or QO, at right angles to LO. If we 
add the deviation which the first lens produces in 
the direction LO io that which the second lens pro- 
duces in this direction, the result is the deviation in 
this direction which the two lenses would produce 
acting in combination ; and this same deviation might 
evidently be produced hy a certain prism whose edge 
is perpendicular to iO, or by a certain cylindrical 
lens whose axis is represented by QO. In the same 
way the deviation produced by the two lenses in the 
direction QO might also be produced by a cjdindrical 
lens whose axis is represented by L 0, 

From this we see that any two cylindrical lenses 
have upon a ray of light the same deviating effect 
as two other lenses whose axes are at right angles. 
This is true for any point on the lens and for any 
position oi LO and QO; but we do not mean that 
the same two lenses at right angles are equivalent 
to the obliquely inclined lenses for different posi- 
tions of the point P, The problem that we wish 
to prove is that for a particular position of the lines 
LO and QO the same two cylindrical lenses are equiv- 
alent to the obliquely inclined lenses irrespective of 
the position of the point P at which the ray meets 
the lens. 

If the focal length of the two lenses were equal, 
we should find the combined action of the two lenses 



154 HANDBOOK OF OPTICS 

in the direction PKhj adding the displacement PE, 
produced by the first lens, to PH^ produced by the 
second lens ; but when the strength of the lenses is 
not the same, these distances do not measure the dis- 
placements which the two lenses produce in the same 
time.* 

Hence we must find an expression from which we 
can reckon the deviating power of the two lenses 
acting through the same distance. Since all rays 
parallel to the axis of the refracting system are re- 
fracted to the focus or focal line of the lens, it is 
evident that the displacement which any ray under- 
goes in the plane of the lens is proportional to the 
distance from the axis at which the ray meets the 
lens ; and so long as the focal length remains un- 
changed, this displacement measures the deviating 
power of a lens at any point on its surface. It is 
also evident that the time required to produce a 
certain displacement varies inversely as the focal 
length of the lens, or directly as the dioptric power. 
Hence the deviating power of a lens at any point 

on its surface is measured by the expression — , in 

which D represents the distance from the axis at 
which the ray meets the lens and F the focal 
length. 

Thus the deviating power of the first lens in the 

* Compare Chap. L, p. 22. 



BI-CYLINDRICAL LEXSES 155 

meridian PM is expressed by —^^ P being any 

F 

point on the lens; and the deviating power of the 

second lens for the same point is expressed by — — ; 

or if we replace the focal lengths by the dioptric 
values, and represent these by C and (7^, respec- 
tively, we have PM - C as the measure of the devi- 
ating power of the first lens, and PN - C^ as the 
measure of this power for the second lens. 

Let the angle AOB^ which is included between 
the axes of the two cylindrical lenses, be represented 
by the letter a ; and let us assume that in a certain 
position of LO and QO the two obliquely inclined 
lenses may be replaced by two other lenses at right 
angles. The unknown angle LOA^ which the axis 
LO must make with OJ., is denoted bv x, and the 
angle LOB, which is equal to a — x,i^ denoted by y. 
It is readily seen from the fignre that P3IE, MPP\ 
and ORK ?iYQ each equal to x, and that PNS, S'PN 
and OSK are each equal to y. 

The dioptric value of the assumed lens whose 
axis is 0^ is denoted by C^^, and that of the lens 
whose axis is LO is denoted by 0^. If the lens O^ 
is equivalent to the combined action of the two 
lenses and 0-^ in the direction iO, we have the 
equation 

PK' 0,^= PE'0 + PH-C^, (1) 



156 HANDBOOK OF OPTICS 

for PK' C^ expresses the deviating power of the 
lens 0^ in its refracting meridian PK^ and PE • 
and PH ' O^ express the deviating power of the 
first and second lens respectively in tlie direction 
PK\ and, since when both lenses are convex tlie 
displacements PE and PH are both toward the 
axis OQ^ we must add the deviating powers of 
the two lenses to find the equivalent lens.* In 
the direction PK' or Q the displacements EM and 
HN are in opposite directions, and the difference in 
power of the two lenses in this direction will express 
the power of the equivalent lens. Hence we have 
also the equation 

PK' . C^ = RN'0^- EM' 0. (2) 

But PE =PM sinx, PH = PJST- sin y, 

HN^PN'Gosy, 'dJid EM=:PM'Gosx. 
Hence equation (1) becomes 

^ PM . ^ , PJV . ^ 

But PM= PR • sin x, and P]Sr= PS • sin y ; 
also, PE = PK- BK, and PS= PK+ KS, 

* We take as the typical case two convex lenses ; the same 
demonstration is applicable if one or both lenses be concave, it 
being only necessary to change the sign of C or of (7i, or of both. 



BI-CYLINDRICAL LENSES 157 

Hence we have 

or, (^2 = sin^ x - C + sin^ U ' ^i 

_||.sia2a;.C + ||.sin2^.(7,. (3) 

If this is true for all positions of P, it must be 

true when P lies on the line LO. It is readih^ seen 

that when P is moved over to K'^ RK and KS 

both become zero, and our equation reduces to the 

form 

C^ = sin2:?; . C+ sin2^ . q^ . (4>^ 

and since C^ is a constant quantity for all positions 
of P, if it is an equivalent lens for and (7j in its 
refracting meridian, then the algebraic sum of the 
last two terms of the second member of equation (3) 
must be equal to zero.* Thus we have as the con- 
dition of equivalence for all points on the lens that 
the expression 

* Bonders and Weiland in their demonstrations have assumed 
that a lens C is in the meridian TK equivalent to another lens 
whose dioptric power is expressed by (7 • sin^ x. This we readily 
see is true only for points on the axis L 0. 



158 HANDBOOK OF OPTICS 

should be equal to zero ; or, 

but we see from the figure that RK=^ OK- cot x, 
and KS — OK - cot i/, Substitutmg and replacing 

the cotangent of these angles by -^, we have 

sin 

C • sin X • cos x= 0-^ sin y • cos 7/ ; 
or, C - sin 2x= (7^ sin 2 y ; 

or, C ' sin 2x= C^ sin 2(a — x). 

By reduction we obtain the equation 

O + C\ cos 2 a 

and from this we know the value which must be 
assigned to x. 

In the same way we find from equation (2), 

^ PN ^ PM ^ 

^3 = -pK' ^°^ ^ ■ ^1 ~ 'PK' ^°^ "^ ■ ' 

but PN= PS' ■ cos ?/, and PS' = PK' + K'S', 
also Pif= PE' ■ cosx, and Pi?' = K'B' - PK'. 



BI-CYLLS'DRICAL LENSES . 159 

Hence 

^ PK' + E'S' 2 ry K'R - PK' „ ^ 
^ " ~PK' ■ y ' ' PK' ' ■ ' 

or, Cq = cos^ y ' ^\ + c^s^ ^ • ^ 

When P lies on the axis OQ, K'S^ and ^^i2^ 
both become zero, and 

Cq = cos^ 1/ ' ^1 + cos^ X • G. (5) 

Hence 

-^— cos2^ . C^ - -^^ cos2:^ . (7=0, 
irom which 



^'i?^ C0s2y. C\' 

or, since ^'aS"^ = OK' • tany, 

and K'R = 0K' i^nx, 

sin ^?/ cos :r _ cos^ x • G ^ 
cos 2/ sin X cos^ 2/ • C^ ' 

or, (7 sin 2 a; = (7^ sin 2 ^Z ; 

from which, as above, 

, o G. sin 2 a 

tan Ix = -— — 1 • 

G + 6\ cos 2 a 



160 HANDBOOK OF OPTICS 

Thus we find that the angle LOA is the same in 
order that C^ should be equivalent to C and C^ in 
the direction 0^ as when C^ is equivalent to these 
lenses in the direction LO^ and therefore it is proved 
that by giving a suitable value to the angle LOA^ 
we may substitute for the obliquely inclined lenses 
two other lenses at right angles, or, the equivalent 
of the latter, a sphero-cjdindrical combination. To 
find the dioptric value of 0^ and C^ we have only to 
substitute the values of x and y which we now know 
in equations (4) and (5). If we add these two 
equations, we see that 

^2 ^" ^3 "^ ^ "I" ^1' 

since sin^ x + cos^ x and sin*^ y + cos^ y 

are each equal to unity. 

In practice it is not necessary to resort to this cal- 
culation, as was stated in the first part of the present 
chapter. The equivalent may be found in the fol- 
lowing manner : Place the two lenses in a trial 
frame with their axes in proper position ; and, hold- 
ing the frame about -^ metre in front of the eye, look 
through the lenses at a test-type card or other rec- 
tangular object distant five metres or more. Rotate 
the frame in the plane of the lenses until the posi- 
tion is found in which there is no angular deviation 
of the vertical and horizontal edges of the card; 



BI-CYLINDRICAL LENSES 161 

then move the frame slightly from right to left or 
vice versa until there is no break in the vertical line 
as seen through the lens and above it. Notice where 
this unbroken line cuts the trial frame, and thus read 
off on the frame the angular marking, which gives 
the position of the axis of one of the equivalent 
lenses ; the axis of the other is at right angles to 
this. Having found the position of the axes, we 
next neutralize the meridian of least refraction by 
means of a spherical lens ; adding now a suitable 
cylindrical lens, with axis in the meridian already 
neutralized, we neutralize the meridian of greatest 
refraction. The sphero-cylindrical combination of 
equal but opposite power to that required for neu- 
tralization represents the equivalent of the two ob- 
liquely inclined lenses. The accuracy of this method 
is limited only by the intervals between lenses in the 
trial case; it is therefore sufficiently accurate for 
practical purposes. The nearest equivalent which 
exists in the trial case can be found in a few 
moments. 

In this way also can the lenticular astigmatism of 
the eye be found.* Suppose, for instance, that the 
entire astigmatism of the eye, as found by subjective 
or objective test, is three dioptres, the meridian of 

* It was for the purpose of showing how to find the lenticular 
astigmatism of the eye that Bonders gave his solution of the bi- 
cylindrical problem. 



162 HANDBOOK OF OPTICS 

least curvature being 60 degrees from tlie horizontal 
line. This can be represented by a convex lens of 
three dioptres, witli its axis at 60 degrees. If the 
corneal astigmatism be found by the ophthalmometer 
to be two dioptres, with the meridian of least cur- 
vature at 130 degrees, this will be represented by 
a convex lens of two dioptres with axis at 130 de- 
grees. The lenticular astigmatism is evidently equal 
to the entire astigmatism less that of the cornea. 
To find this we place in the trial frame a convex 
cylindrical lens of three dioptres with axis at 60 
degrees, which represents the entire astigmatism. 
If we neutralize the corneal astigmatism, we have 
remaining the lenticular astigmatism. The corneal 
portion is neutralized by a concave lens of two 
dioptres wdth axis at 130 degrees. Placing this lens 
in the trial frame, we have a combination of two 
cylindrical lenses obliquely inclined, and the astig- 
matism of this combination represents the lenticular 
astigmatism of the eye. By neutralizing the combi- 
nation, we derive the lens equivalent of the lentic- 
ular astigmatism and the angle of its axis. 



CHAPTER XI 

OBLIQUE REFRACTION THROUGH LENSES 

In our study of refraction we have considered 
only pencils of liglit whose central ray or axis is 
perpendicular to the refracting surface. Such a 
pencil is called direct. We have also learned that 
a spherical surface has greater refractive power for 
rays Avhich meet it at a distance from the axis than 
for those which pass near the axis. We have been 
obliged in order to escape spherical aberration to 
limit our consideration to those rays which do not 
deviate far from the axis. We are justified in this, 
since only small pencils can enter the eje. 

We shall now investigate the refraction of small 
pencils, the axis of which is not perpendicular to the 
refracting surface. Such a pencil is called oblique. 
Refraction by oblique pencils takes place when we 
look through a tilted lens, as is frequently done in 
the use of spectacles. The complete analysis of this 
subject is difficult and requires a knowledge of the 
higher mathematics, but we can study oblique 
refraction in an elementary manner, and this will 
enable us to understand the problems which present 

163 



164 HANDBOOK OF OPTICS 

themselves in ophthalmology. A complete solution, 
so far as relates to tilted spectacle lenses, has been 
worked out by Dr. John Green of St. Louis ; ^ and 
a general solution of the problem of oblique spher- 
ical refraction is given in Heath's " Geometrical 
Optics." f From the formulae derived by these 
investigations, the exact value of a lens when tilted 
at any angle can be found. We shall suppose the 
lens to be tilted only in the meridian of vertical 



Fig. 43. 

refraction. Let A (Fig. 43) represent the princi- 
pal point of a refracting surface. The direct pencil, 
whose axis is PA and whose peripheral ray is PT^ 
will after refraction meet the axis at Q, Let 
RPR^ represent a pencil meeting the surface 
obliquely, PR being the axis of this pencil. We 
have learned from our previous studies that it is 
sufficient to determine the deviation of light in two 
meridians at right angles to each other. We shall 

* Trans. American Ophth. Soc.^ 1890. 

t Heath's "Geom. Optics," 2d ed., p. 179. 



OBLIQUE REFRACTION THROUGH LENSES 165 

therefore investigate the refraction of the oblique 
pencil first in the vertical meridian or plane of tlie 
paper, as represented by B! B!\ and then in the 
horizontal meridian, as represented by SS^ , We 
may consider the rays PBJ ^ PR^ and PPJ' as rays 
of the direct pencil which are so far from the axis 
that spherical aberration cannot be neglected. 
These rays will after refraction meet the axis at Q^, 
Q^, and ^3, respectively. The refracted rays R Q^ 
and RQ2 will meet at q\ RQ^ and R^^Q^ will meet 
at qy If the pencil be small, q and q^ will be so 
near to each other that we may regard them as 
identical. Then the point of intersection q will 
be the focus of the pencil in the vertical meridian, 
and Rq will be the focal distance in this meridian. 
If now we take the raj^s PS and PS' in the hori- 
zontal meridian, it is evident that after refraction 
they will meet the axis at Q^ ; for their vertical 
distance from A is the same as that of the ray PR^ 
the axis of the pencil. Hence Q^^ is the horizontal 
focus, and RQ2 is the horizontal focal distance. 
From this we see that oblique spherical refraction 
is astigmatic. Since Rq is the focal distance in 
any vertical meridian, all rays of the pencil must 
intersect a straight line passing through q and par- 
allel to SS',^ Likewise all rays must have their 

* SS', being a small arc, does not materially differ from a 
straight line. 



166 HANDBOOK OF OPTICS 

horizontal intersections on the line ^1^3; and since 
the pencil is small, the error will be inappreciable 
if we replace ^^^3 by a line through Q^ parallel to 
WR\ These lines drawn through q and Q^ are 
the focal lines of the pencil, and qQ^ is the focal 
interval. It is clear that the focal interval increases 
as the distance of the oblique pencil from the axis 
PA increases. As AQ is the focal distance for the 
direct pencil, it is seen that by the change from 
direct to oblique refraction the focal distance is 



\ Fig. 44. 

shortened in both vertical and horizontal meridians, 
but more in the former than in the latter; for AQ^ 
and Rq are both less than AQ, and i^g^ is less than 
AQ^, In the question before us we have oblique 
refraction at two surfaces. We shall consider only 
the case in Avhich the axis of the pencil passes 
through the optical centre of the lens, so that after 
the two refractions its direction is parallel to that 
before refraction ; and we shall disregard the lateral 
displacement due to the thickness of the lens. 

In Fig. 44 let BJPB!^ represent the vertical 
section of a pencil meeting the lens obliquely at 



OBLIQUE REFRACTION THROUGH LENSES 167 

R' R^ ; then since the axis of the pencil, PR^ passes 
through the optical centre of the lens, its direction 
after emerging from the lens will be parallel to PR, 
Let Q be conjugate to P in the vertical meridian. 
We have seen the refractive power of the first sur- 
face is increased by tilting the lens; consequently 
a shorter incident pencil R'PR^ will cause the 
refracted pencil to assume tiie convergence R^qR^' 
than if the pencil were direct. Likewise if we 
suppose a pencil T' QT" to proceed from Q so that 
after refraction it assumes the divergence RqW^^ 
appearing to proceed from ^, the pencil T' QT" will 
be shorter than if it were direct. Since the path of 
light is reversible, the same will be true when 
T^ QT'^ is an emergent pencil; in other words both 
the incident and emergent pencils are shortened by 
tilting the lens, and the lens is increased in refrac- 
tive power. The same reasoning applies also to 
the refraction in the horizontal meridian, but in 
this meridian the increase of power is less at each 
surface than in the vertical meridian. Hence a 
spherical lens tilted in its vertical refracting me- 
ridian is equivalent to a sphero-cylindrical lens. 
Reference to Fig. 43 will show that as the tilting 
is increased, the amount of astigmatism increases 
very rapidly. 

If a cylindrical lens whose axis is horizontal be 
tilted in its vertical refracting meridian, the increase 



168 HANDBOOK OF OPTICS 

in power will be tlie same as that of a similar spher- 
ical lens in the vertical meridian. If the axis of the 
lens be vertical and it be tilted in this meridian, the 
increase in power will be the same as that which a 
spherical lens undergoes in the horizontal meridian 
when tilted in the vertical meridian. 

We should learn two practical points from this 
study : first, the necessity of giving the proper 
inclination to spectacles, especially when strong 
lenses are used ; and, secondly, the uncertainty of 
effect of very weak cylinders in combination with 
strong spherical lenses. A spherical lens of four 
dioptres acquires more than one-quarter of a dioptre 
of astigmatism by tilting it through an angle of 15 
degrees. Hence a convex cylindrical lens of .25 D., 
axis vertical, combined with a convex spherical lens 
of 4 D., would be completely nullified by a slight 
amount of tilting ; on the other hand, if the axis 
of the lens be horizontal, its action will be practi- 
cally doubled by the same tilting. 



CHAPTER XII 

THE EFFECT OF PRISMATIC GLASSES UPON 
RETINAL IMAGES 

In Chapter I. we considered refraction of rays 
through prisms. We must now investigate in an 
elementary way the more difficult subject of refrac- 
tion of pencils of light such as enter the eye. The 
study of the effect of prisms upon stereoscopic vision 
belongs to physiology, and is discussed in treatises 
on physiological optics. We shall confine our atten- 
tion to the influence of prisms upon the size and form 
of retinal images. 

For convenience we repeat the following : 

1. A principal plane of a prism is a plane jDcrpen- 
dicular to the edge of the prism, and therefore to 
each face of the prism.* 

2. By the lav/ of refraction, the incident and re- 
fracted raj'S and the normal to the surface all lie in 
the same plane. 

3. From the equation sin i = n • sin r, it follows 
that the greater the angle of incidence, the greater is 

* The principal plane of a prism bears no analogy to the princi- 
pal plane of a spherical refracting surface. 

169 



170 



HANDBOOK OF OPTICS 



the deviation of the ray ; and the greater the angle 
of incidence, the greater is the increase in deviation 
for a fixed increase in the angle of incidence, the rate 
of change in deviation increasing very rapidly as the 
angle approaches 90 degrees. 

From (1) and (2) we see that a ray which enters 
the prism in a principal plane must lie in this plane 
after emergence. 

A ray which enters in any other plane passes out 
of the prism in a plane parallel to that in which it 




Fig. 45 



entered, the amount of separation between the planes 
depending upon the thickness of the prism ; and the 
deviation which this ray undergoes in the principal 
plane is greater than if the ray were in this plane.* 

The effect which these facts have upon images will 
appear subsequently ; we must first examine the re- 
lation between the length of the pencil and the size 
of the image. Helmholtz' formula applies here as in 
all cases, but this relation is indicated clearly in Fig. 
45. Let AP represent the linear dimension of an 



* For the demonstration of this see Appendix II. 



PRISMATIC GLASSES 171 

object, then ^aS' represents its image on the retina. 
The distance from P to the cornea is the length of 
the pencil before refraction, RPO is the angle of 
divergence of the pencil, and A OP or QOS is the 
visual angle which the object snbtends. 

Thus we see that as the incident pencil becomes 
shorter the image on the retina increases. If, by any 
means, rays of light from an object AP are rendered 
more divergent, so that the pencils appear to proceed 
from A^P\ the image will be enlarged from QS to QS\'^ 

The effect of prisms upon the length of pencils 
varies greatly with position of the prism as regards 
the light which passes into it. If the prism be of 
small refracting angle, all rays which pass through it 
near the position of minimum deviation will undergo 
practically the same amount of deviation, f Small 
pencils passing through such a prism in this position 
undergo no change in length, for if the rays all have 
the same amount of deviation their relative diver- 
gence will be unaltered in their passage through the 
prism. But all other pencils will be altered in 
length ; and to prove this let Fig. 46 represent a 
principal section of a prism. From draw It 0R\ 
representing a pencil in the position of minimum 
deviation, and also SOS^ and TOT\ two pencils, the 

* We neglect as inappreciable the slight change in position of the 
optical centre 0, due to change in refractive state of the eye upon 
approximation of the object. t Chap. I., p. 21. 



172 HANDBOOK OF OPTICS 

former being nearer the apex and the latter nearer 
the base of the prism than ROR'. As has been 
stated, tlie pencil ROR undergoes no change in 
length. By the first refraction the pencil SOS' is 
increased in length, for OS makes a greater angle 
with the normal to the surface than does OS' ; its 
deviation is consequently greater and the divergence 
of the pencil is diminished. By the second refrac- 
tion the divergence of the pencil is for a similar 

'^"^-^.^ 




Fig. 46. 



reason increased ; but since the angles of refraction 
are greater at the first surface, the difference in the 
amount of deviation undergone by OS and OS' is 
greater at this than at the second surface ; conse- 
quently the lengthening of the first surface out- 
balances the shortening at the second surface. In 
the pencil TOT' the angles of refraction are greater 
at the second surface, and the shortening at this sur- 
face outbalances the lengthening at the first surface. 
Furthermore it will be observed that pencils proceed- 



PRISMATIC GLASSES 173 

ing from a point will not, after passing through a 
prism, appear to come from a point ; but if the axis 
of the pencil lie in a principal plane of the prism, as 
we suppose in Fig. 46, rays proceeding from will 
appear to come from two focal lines parallel and per- 
pendicular, respectively, to the edge of the prism. 
This follows from the analogy to astigmatic refrac- 
tion at cylindrical surfaces ; for the rays of the 
pencil will not be deviated in the direction of the 
edge of the prism, the alteration in length taking 
place in the meridian at right angles to this edge. 
Since we deal only with small pencils, we may for 
our present purpose ignore the astigmatic effect and 
regard the pencil SOS' as proceeding from a point 0'^ 
after refraction by the prism. An object seen by 
pencils such as SOS^ appears minified in the direc- 
tion of the principal plane, since the incident pencils 
as received by the observer's eye are lengthened ; on 
the other hand, an object seen by pencils such as 
TOT appears magnified in this direction, since the 
incident pencils are shortened. 

From this we see that as the apex of a prism is 
turned toward the object, the image is magnified in 
the principal plane of the prism ; and as the base of 
the prism is turned toward the object, the image is 
minified in this plane.* 

* Upon this principle is constructed a toy, — the "laughing 
camera," — which is sold on the streets and in the shops. 



1T4 



HANDBOOK OF OPTICS 



Let us now study the effect of a prism placed 
before the eye with its principal plane horizontal. 
If the prism have a refracting angle of 15 or 18 de- 
grees, objects in the field of view will undergo 
marked distortion. This is due not only to the 
actual distortion of the retinal image, but also, in 
part, to mental impressions of previous experience. 



(a) 









Fig. 47. 

When prisms are worn before both eyes there is still 
further confusion arising from altered convergence 
and perspective.* 

In Fig. 47 (a) represents a square object so placed 
that 0, its middle point, is seen by rays passing 
through the prism with minimum deviation ; then 
if the base of the prism is placed to the right, (J) is 
a representation of the object as it appears on the 



* Articles bearing upon this subject have been published by a 
number of writers. Reference may be made to the following : 
Helmholtz' "Physiological Optics"; "Stereoscopic Illusions 
Evoked by Prismatic and Cylindrical Spectacle Glasses," John 
Green, Trans. American Ophth. Soc.^ 1889. 



PRISMATIC GLASSES 175 

retina. The portion of the figure near is un- 
affected by the prism, but the portion toward the 
base of the prism is minified, and that toward the 
apex is mag]iified in the liorizontal meridian. The 
point A is displaced toward 0, and any other point 
of the line AE is displaced in this direction to a 
greater extent than is A^ because a ray not in a 
principal plane is more deflected than a correspond- 
ing ray Ij'ing in this plane. As this increase in 
deviation continues at an increasing rate with the 
increase of the distance of the point from A^ the 
line AE appears curved. For the same reason ON 
and BF appear curved. 

Neglecting, as we do, the thickness of the prism, 
there is no displacement at right angles AE^ and 
therefore EF is unchanged in direction in the retinal 
image. Nevertheless EF does seem to be changed 
in such manner as to assume the form represented 
in Qc) — a change which is due to illusion. The 
line AE^ as seen through the prism, appears at an 
increased distance Avhile its image is unchanged ; 
consequently the line seems to be increased in length. 
For a similar reason BF appears shortened. This 
increasing apparent distance, as the attention is 
directed farther to the left, also causes this part of 
the figure to assume a convex cylindrical form ; 
while that portion toward the extreme right appears 
concave. This effect can be obtained by looking 



176 HANDBOOK OF OPTICS 

through a strong prism at the opposite wall of a 
room, turning the prism so as to get the extreme 
magnifying and minifying power. Since the lower 
half of the figure is a reproduction of the upper half, 
it needs no explanation. 

Similar distortions and illusions must be produced 
by weak prisms, though in a less marked degree ; 
it is not surprising therefore that unpleasant sen- 
sations arise when persons attempt to wear com- 
paratively strong prismatic glasses as spectacles. In 
many cases, however, this disturbance passes away 
after the glasses have been worn for a short time. 



CHAPTER XIII 

THE KEFLEXION OF LIGHT 

The law of reflexion — namely, the angles of inci- 
dence and reflexion lie in the same plane and are 
equal — was, as we have stated, known to the 
ancients. 

As we have found the law of refraction to follow 
as a necessary consequence of the wave theory of 
light, being due to the varying velocity of light in 
different media, so also the law of reflexion cor- 
roborates this theory. In reflexion the light does 
not pass out of the first medium ; its direction, 
however, is reversed. Hence, if we make n equal 
to minus 1 in any equation pertaining to refraction, 
we ought to get the corresponding equation for 
reflexion. By making this substitution we arrive 
at identical formulae with those obtained from in- 
dependent geometrical construction. 

The phenomena of reflexion, so far as they con- 
cern the ophthalmologist, are exceedingly simple. 

The illumination of the interior of the eye is ac- 
complished by means of a mirror, which reflects light 
from a flame, while the observer is so situated as to 

177 



178 



HANDBOOK OF OPTICS 



be in the patli of the light returning by reflexion 
and refraction. 

If a plane mirror is used, light is thrown into the 
eye by pencils which appear to come from behind 
the mirror and from a distance equal to that of the 




Fig. 48. 



lamp from the mirror. This is illustrated in Fig. 48. 
It can easily be proved that the angles A and A^ are 
equal. 

If a concave mirror is used, the light appears to 
come from a point in front of the mirror. This we 
deduce from equation (a), page 30. If we make n 
equal to minus 1 in this equation, the result is an 
equation expressing the relation between the conju- 



THE REFLEXION OE LIGHT 179 

gate foci after reflexion at a spherical surface. Mak- 
ing this substitution, equation (a) becomes 

1_1__2 1_2 , 1 

f is the distance of the flame from the mirror, /' is 
the distance of its conjugate from the mirror, and r 
is the radius of curvature. Since the mirror is con- 
cave, r is negative. Hence, / being positive, /' is 

2 1 

negative when - is greater than - and positive when 

- is less than -. 
r f 

In other words, if / is greater than -, the two 

conjugate foci lie on the same side of the mirror ; 
if / is less than -, both / and f^ are positive, a]id 

consequently lie on opposite sides of the mirror. 

The radius of curvature of the concave mirrors 
used in ophthalmoscopy does not exceed ^ metre ; 
and as the distance of the flame from the mirror is 
greater than half this radius, the point from which 
light proceeds is in front of the mirror. The 
illumination of the eye bv the concave mirror is 
therefore more intense than that by the plane 
mirror. 

Figure 49 illustrates reflexion by the concave 
mirror ; light from A is focused at A' . HA rep- 



180 HANDBOOK OF OPTICS 

resent a gas flame, an image of this flame will be 
formed at A^. 

Continuing the study of reflexion at a spherical 
surface, we see that if we make / equal to infinity, 

f is equal to -, that is, the principal focal distance 
of a spherical mirror is equal to one-half of tlie 

radius of curvature.* If the mirror is concave, - 

2 

is negative, and the principal focus is half way be- 




FiG. 49. 



tween the centre of curvature and the surface of the 
mirror. If the mirror is convex, - is positive, and 

the principal focus lies behind the mirror. Clearly 
such a focus must be virtual, while the negative 
focus of the concave mirror is real. Hence if we 
use equation (a), real foci in reflexion are negative 
while virtual ones are positive. 

*It will be observed that in reflexion the two principal foci 
coincide. 



THE REFLEXION OF LIGHT 181 

It must be borne in mind that the equation ap- 
plies only to small pencils near the axis of the sur- 
face, for spherical aberration occurs in reflexion as 
in refraction. 

The same relation exists between the size of object 
and image as in refraction, viz. : 

__ u 

If u be positive, that is, if the object be without 
the principal focus, as in Fig. 49, then the image 
will be positive or negative according as F is posi- 
tive or negative. When F is negative, as in the 
concave mirror, the image is real and inverted; 
when F is positive, as in the convex mirror, the 
image is virtual and erect. 

In the concave mirror u is negative Avhen the ob- 
ject lies nearer the mirror than the princijDal focus ; 
in the convex mirror u cannot be negative, since 
the principal focus is virtual and behind the mirror. 
Hence in reflexion at concave mirrors the image is 
real and inverted when the object lies without the 
principal focus ; virtual and erect when tiie object 
lies within this focus. In reflexion at convex 
mirrors the image is always virtual and erect. 

XT 

Since the size of the image is equal to o - —, the 

u 

virtual image formed by the concave mirror is larger 



182 HANDBOOK OF OPTICS 

than the object, for u^ lyhig between F and the 
mirror, must be less than F. 

The real image formed by the concave mirror is 
less than the object when u is greater than F^ and 
vice versa. 

The erect virtual image formed by the convex 
mirror is always smaller than the object, since u is 
necessarily greater than F. 

Since all refracting surfaces act also as reflecting 
surfaces, the three refracting surfaces of the eye, 
namely, the cornea, the anterior, and the posterior 
surface of the lens, must furnish also three reflect- 
ing surfaces. Consequently, when an object is held 
before the eye, there must be formed by reflexion 
three images of the object. The first, formed at 
the convex surface of the cornea, is virtual and 
erect ; the second, formed at the anterior surface of 
the lens, is also virtual and erect; while the third, 
formed at the concave posterior surface of the lens, 
is real and inverted. These images, as seen with 
the aid of a lighted candle in a darkened room, 
are of great interest to ophthalmologists. The 
formation of all three images is conclusive evidence 
of the presence of the crystalline lens in the eye. 
Furthermore, by the change in relative size of 
these images, the increase in curvature of the lens 
during the act of accommodation can be demon- 
strated. 



THE REFLEXION OF LIGHT Igg 

Since the size of the ima^e varies with — , and F 

is equal to -, we have a means of measuring the 

curvature of the refracting surfaces of the eye, 
provided we can measure the size of the reflected 
images. A brief description of the way in which 
this can be done will be given in the next chapter. 



CHAPTER XIV 

THE OPTICAL PRINCIPLES OF OPHTHALMOMETRY 
AND OF OPHTHALMOSCOPY 

To the genius of Helmholtz we owe the invention 
of the ophthalmometer, an instrument of great pre- 
cision for measuring the curvature of the refracting 
surfaces of the eye. In the construction of this 
instrument Helmholtz employed a device already in 
use by astronomers for the measurement of the stars, 
namely, the production of double images of a single 
object. 

This is possible by means of several contrivances. 
The simplest is the double prism, such as the Mad- 
dox prism found in the oculists' trial case. Helm- 
holtz' device consisted of two plates of glass of 
equal thickness inclined at an angle, as is shown in 
Fig. 50. A pencil of light from o meets the plate 
i), and is refracted as in the figure. The rays, after 
emergence, are parallel to their direction before 
entering the plate, but they undergo a lateral dis- 
placement due to the thickness of the glass, so that 
they all appear to come from A, Likewise, that 
part of the pencil which passes through the plate 

184 



OPHTHALMOMETRY AND OPHTHALMOSCOPY 185 

JE appears to come from B, Hence, if o represent 
a small object, there will appear after refraction 
through the plates two similar objects at A and B^ 
respectively. 





D 




a' 




H/^'^ 


L^^^=^^ 


^3 J 


____— — ■ — " 


^^^"^^ 


-^^^^lf^$ 


^^^^^^^^^:^=^ 


u^^^^^ 


^^LJ 


' — — ~-__^ 


_^^ 




^^^^"^^"""^^ 




^eY 


-^ 


B' 


M 


^\_7 



Fig. 50. 



If a convex lens be placed at Z, a real image of 
A will be formed at A^^ and an image of B will 
be formed at 5'. A second convex lens, Jf, whose 
principal focal plane is A^B\ will render rays from 
the image A^B' parallel ; and the 
image will be focused on the retina 
without accommodation. 

If the circle whose centre is (Fig. 
51) be viewed through a double prism, 
or through two inclined plates ; and, 
if the double images are separated to 
such an extent that the two circles A 
and B appear to touch at 0, it is clear 
that the amount of separation of the 
iraages will be equal to the diame- 
ter of the circle 0, It is also clear that if the 
angle between the two plates can be varied, then, by 
changing this angle to the proper degree, the two 




y 



Fig. 51. 



186 HANDBOOK OF OPTICS 

images can be made to touch as in the figure. 
Knowing the angle between the plates, the thick- 
ness and refractive index of the glass, the amount 
of displacement from to A and from to B can 
be calculated. 

If the object at be a reflected image as seen in 
the cornea, we can obtain its size from the data 
above mentioned. The size of the object, the size 
of the image, and the distance of the object from 
the cornea being known, we deduce the curvature of 
the cornea from the equation, 

_ u _2u 
i F r ' 

The greater the curvature of the surface, the 
smaller is the image ; consequently, if the two im- 
ages touch in the meridian of least curvature of an 
astigmatic cornea, they Avill be separated by an 
interval in the meridian of greatest curvature ; 
while, if they touch in the meridian of greatest cur- 
vature, they will overlap in that of least curvature. 

The construction of the modern ophthalmometer 
of Javal and Schiotz is somewhat different from that 
of Helmholtz ; but the essential optical principles 
are the same in both instruments. 

By the aid of ingenious mechanical devices, ob- 
servations of the corneal curvature have become a 
matter of the greatest ease. In the instrument of 



OPHTHALMOMETRY AND OPHTHALMOSCOPY 187 

Javal and Schiotz the glass plates are replaced by a 
Wollaston prism, which, like the plates, produces 
two images of a single object. 

Certain crystalline substances possess the peculiar 
property of double refraction. Iceland spar is a 
familiar example of a double refracting substance. 
Part of the light entering this material undergoes 
refraction in the ordinar}^ wa}^ while a part possesses 
the property of having different velocities, and hence 
different refractive indices in different directions. 

This is due to the fact that the constitution of the 
substance is such as to offer unequal resistance to 
the passage of light in different directions or axes. 
The tirst portion of light which follows tlie ordinary 
law of refraction is called the ordinary ray ; tlie 
second portion, whose index varies for different 
meridians, is called the extra-ordinary ray. 

If we take a piece of double refracting substance, 
as Iceland spar or quartz, and through it look at an 
object placed in such position as regards incident 
light that the difference in index between the ordi- 
nary and extra-ordinary rays causes a separation of 
these rays, a double image of the object will be 
formed. Wollaston's prism is based upon this 
principle.* 

* For a detailed account of the phenomenon of donble refrac- 
tion and the construction of Wollaston's prism, consult Preston's 
" Theory of Light," or other complete treatise on optics. 



188 HANDBOOK OF OPTICS 

In the modern ophthalmometer, in which the Wol- 
laston prism is used, the objective L is composed of 
two lenses separated by an interval ; and the prism 
is placed in this interval between the lenses. 

The exact amount of separation which the prism 
produces in its fixed position is known. The di- 
ameter of the object from which light is reflected to 
the cornea can be varied at will. This object con- 
sists of two sets of white enamelled disks called mires, 
equally distant from the centre of a connecting bar. 
By increasing or diminishing the distance between 
the mires their separation may be made such that the 
double images are tangent to one another. From a 
scale which has been constructed by previous calcula- 
tion, the curvature of the cornea or its refractive 
power in dioptres can be read off on the bar separat- 
ing the mires. 

The ophthalmoscope is a contrivance by which the 
observer reflects light into an eye, while he is in 
such position as to receive in his own eye the light 
which returns by reflexion and refraction from the 
observed eye. As Helmholtz invented the ophthal- 
mometer, so to him are we indebted for the gift of 
the ophthalmoscope. Prior to this invention, in 
1851, the question of seeing the fundus of the eye 
had attracted much attention. It was of course 
known that the eyes of some animals emit a reddish 
or greenish tint under certain circumstances, and 



OPHTHALMOMETRY AND OPHTHALMOSCOPY 189 

many absurd speculations were indulged in for the 
explanation of this phenomenon. Briicke made a 
thorough study of this subject, and in 1847 gave the 
true explanation.* Indeed, he came so near to the 
invention of the ophthalmoscope as to place in a 
flame an iron tube, through which lie could see the 
fundus of the eye. It was also known prior to these 
discoveries that the fundus of the ej^e would become 
visible if the eye were immersed in Avater. The ex- 
planation of this is similar to that of the glow of a 
cat's eye. 

The ophthalmoscope in its simjDlest form is a plane 
or concave mirror, havinof in its centre a small circu- 
lar opening through whicli the observer receives tlie 
light returning from the observed eye. Reference to 
Fig. 62 shows why the fundus of an eye cannot be 
seen without a special contrivance. If light be re- 
flected into the eve from a flame A alono^ the axis 
AJE^ and along other axes near AU, a small image 
of the flame will be formed on the retina, and at the 
same time a small portion of the fundus adjacent to 
the image will be illuminated by irregular reflexion. 
Light from this portion of the fundus passes out of 
the eye. If the eye is emmetropic, the pencil from 
U, having as its axis UA, is after refraction changed 
to parallel rays ; similarly, pencils from other points 

* Mueller's " Archiv ftir Anat. und Phys.," 1845, S. 387; and 
1847, SS. 225, 479. 



190 



HANDBOOK OF OPTICS 



:z 



Fig. 52. 



near E will have as axes lines ly- 
ing very near EA, and these rays 
will also, after refraction, be par- 
allel to their axes ; thus it is evi- 
dent that the light which returns 
from the examined eye cannot 
within a short distance from the 
eye deviate far from the axis EA. 
Hence we see that without a spe- 
cial contrivance the observer's 
head would necessarily cut off the 
light which illuminates the qjq. 

In the hyperopic eye pencils of 
light from any point on the fundus 
are divergent after leaving the eye. 
If the fundus be at H^ then BHC 
will represent a pencil from H, If 
the rays diverge considerably it 
will be possible for an observer 
to place his head in position to 
receive some of the rays^ and yet 
not obstruct those from the illu- 
minating source. It is on this 
account that a cat's eye glows in 
the dark when the observer is not 
far from the path of the rays which 
enter the eye. Placing the eye 
under water has the same effect 



• OPHTHALMOMETRY AND OPHTHALMOSCOPY 191 

upon pencils as hyperopia, for the refractive index 
of water is nearly the same as that of the aqueous, 
and, the external surface of the water being plane, 
we have in the immersed eye a high degree curva- 
ture hyperopia. 

By referring to Fig. 62 we see that when the fun- 
dus is conjugate to the position from which the illumi- 
nation proceeds, only a small portion of the fundus 
is illuminated, and that the more remote the fundus 
is from this conjugate position, the greater is the 
portion of fundus illuminated. This is the prin- 
ciple upon which is based the method of examina- 
tion known as skiascopy. If the examination be 
conducted with a plane mirror so placed that light 
enters the eye in pencils diverging from a distance 
of one metre in front of the eye, and if the far point 
of the eye be also distant one metre, that is, if the eye 
have one dioptre of myopia, a very small part of the 
fundus will be illuminated. Hence, if the mirror be 
slightly tilted, the area of illumination will at once 
be thrown out of the line of vision of the observer. 
If the eye be hyperopic, then as the mirror is tilted 
the area of illumination will move in the same direc- 
tion, but it will not pass out of view so rapidly, and 
we can observe the motion of the reflex and its at- 
tendant shadow as they move across the pupil. 
When the mirror is so tilted that the light which 
enters the eye appears to come from i, then a 



192 HANDBOOK OF OPTICS 

straight line, drawn through the optical centre of 
the eye, connecting i, H'^ E^ ^ N' ^ and M' ^ replaces 
the axis AM\ hence, no matter what may be the 
refractive condition of the eye, the area of illumina- 
tion always moves in the direction of the tilting of 
the mirror. But in myopia such that the fundus is 
at M, beyond the conjugate iV, there will be formed 
to the left of A an aerial image of the illuminated 
area, and this will evidently move in the opposite 
direction to the tilting of the mirror. If the ob- 
server is farther from the eye than this image, he 
must therefore see the reflex and shadow move in 
the opposite direction to the tilting. 

By observing the movement of the shadow in dif- 
ferent meridians of the eye, the test can also be used 
for the detection of astigmatism. 

If a concave mirror be used so that light enters 
the eye diverging from a real image in front of the 
mirror, the movements of the shadow will be oppo- 
site to those which occur with the use of the plane 
mirror ; the reason for this is apparent. 

With such accuracy can the motion of the shadow 
be observed that this method, in skilful hands, sur- 
passes all other objective means of examining the 
refractive condition of the eye. 

Since light leaves the emmetropic eye in parallel 
rays and the hyperopic eye in divergent pencils, 
these rays may be brought to a focus on the retina 



OPHTHALMOMETRY AND OPHTHALMOSCOPY 193 

of the observer, thus forming an image of the illumi- 
nated part of the fundus under examination, so that 
in both hyperopia and emmetropia the details of the 
fundus can be seen, provided the observer is suffi- 
ciently near the examined eye to receive light from 
an appreciable area of the fundus. Moreover, since 
in hyperopia a larger part of the fundus is illumi- 
nated than in emmetropia, the details can be seen at 
a greater distance in hyperopic eyes. 

The rays from hyperopic and emmetropic eyes, 
being either divergent or parallel, will never meet 
in a real aerial image ; hence the image, as seen by 
the examiner, will always be erect. The examina- 
tion of the erect image is called the direct method of 
ophthalmoscopy. 

Light emerging from a myopic eye is convergent, 
and if the eye of an emmetropic observer be nearer 
the e5"e than the far point, a concave lens must be 
used to see clearly the details of the fundus. The 
image as thus seen will be erect ; but at the far 
point of the eye, which we know is conjugate to the 
retina, an inverted aerial image will be formed, and 
from this, divergent pencils will enter an observer's 
eye situated beyond the image. A concave lens will 
be no longer required ; on the other hand, exercise 
of the accommodation or a convex lens varying with 
the distance of the eye from the image is necessary. 
The examination of the inverted image is called the 



194 HANDBOOK OF OPTICS 

indirect method of ophthalmoscopy. As thus de- 
scribed, this method would be practicable only in 
highly myopic eyes ; but we may produce the aerial 
image in all cases by holding a strong convex lens 
in front of the eye to be examined. The stronger 
the lens the larger will be the field of view and 
the smaller the image ; hence the strength of the 
converging lens may be varied to suit the purpose of 
the examiner. 

Finally let us investigate the apparent size of the 
optic disk — the most conspicuous object revealed 
by the ophthalmoscope — as affected by the various 
refractive conditions of the eye. We shall first con- 
sider the examination by the direct method. When 
the eye under examination is emmetropic, no change 
in apparent size is produced by varying the distance 
between the observed and observer's eyes ; for we 
see from Fig. 50 (in which A^ B^ may represent the 
disk, the lens M may represent the observed, and HN 
the observer's eye respectively) that the visual angle 
HON under which the disk is seen does not vary 
with the distance between the two eyes.* The field 
of view becomes smaller as this distance increases, for 
more rays pass outside of the eye ! but no change in 
size is produced. If the observed eye be hyperopic, 
rays after leaving it will be divergent, and this same 
divergence might be caused by placing a suitable 

* See also Fig. 45, p. 170. 



OPHTHALMOMETRY AND OPHTHALMOSCOPY 195 

concave lens in contact with an emmetropic eye. A 
lens in this position is without the anterior focus of 
the observer's eye. We know that a concave lens 
placed without the anterior focus of an optical sys- 
tem diminishes the size of images formed by it, and 
that the minifying effect increases as the distance of 
the lens from the focus. Hence in hyperopia the 
disk appears smaller than in emmetropia, and its 
apparent size diminishes as the distance between 
the eyes increases. In myopia the pencils are con- 
vergent after leaving the eje, just as if a convex lens 
were placed before the eye ; thus the disk in this 
case appears larger than in emmetropia, and the 
apparent size increases with the distance between 
the eyes. 

In astigmatism the disk appears smaller than 
normal in the meridian of hyperopic and larger than 
normal in the meridian of myopic refraction. Hence 
supposing the disk to be circular in form, it appears 
oval in astigmatism witli the long axis in the merid- 
ian of greatest refraction. 

In the indirect method of examination the image 
as seen by the observer varies in size with the aerial 
image formed by the convergent lens ; hence we 
must investigate the size of this image as affected by 
the refractive condition of the examined eye. In 
hyperopia the concave lens, which we suppose to be 
placed in contact with the cornea of a normal eye, 



196 HANDBOOK OF OPTICS 

has a magnifying or minifying effect upon the aerial 
image according as the distance of the lens from the 
eye is less or greater than the focal length of the 
lens. Therefore in the indirect examination the disk 
appears larger than normal when the distance of the 
lens from the examined eye is less than the focal 
length of the lens ; when this distance is equal to 
the focal length of the lens, the disk appears nor- 
mal in size, and when the distance of the lens from 
the eye is greater than the focal length of the former, 
the disk appears smaller than normal. 

In myopia we have the opposite conditions, that is, 
the disk appears diminished in size when the distance 
between the eye and lens is less than the focal length 
of the latter ; and, increasing with the distance of 
the lens, the disk appears larger than normal when 
the distance of the lens from the eye is greater 
than the focal length of the lens. 

Thus also in astigmatism the disk undergoes the 
opposite distortion to that which occurs in the direct 
examination, provided that the distance of the lens 
from the eye is less than the focal length of the lens, 
and when the lens is farther from the eye than this 
length the distortion is the same in both methods. 

This distortion, first described by Knapp, was 
formerly used as a test for astigmatism. By placing 
a suitable lens before the eye the distortion can be 
made to disappear, and this lens represents the 



OPHTHALMOMETRY AND OPHTHALMOSCOPY 197 

amount of astigmatism present. While this test 
has given way to other more delicate ones, the 
phenomenon is, in high degrees of ametropia, so 
striking that it must attract the attention of every 
student of ophthalmoscopy. 



APPENDIX I 



For the convenience of those who may not be familiar 
with trigonometrical formulae, we append the following 
synopsis : 

In the figure, ABODE represents a quadrant, or 90 
degrees of the circumference of a circle, of which AB, 

BC, CD, and DE are equal 
arcs. The angles AOB,^ 
BOC, COD, and DOE are 
also equal. 

If BF be drawn perpen- 




dicular to OA, then 



BF 
OB 



IS 



called the sine of the angle 
AOB; similarly, — — is the 



Fig. 53. 



sine of COR. 



BF . 



00 

OF 

OB 
OF 
BF 



is called 



the cosine of AOB, ^ is the tangent, and — - the cotan- 

OF Bli 

gent of this angle. 

The angle xWC is twice the angle AOB, but it is 
readily seen that the perpendicular CH is less than 
twice the perpendicular BF] hence the sine of twice an 
angle is less than twice the sine of the angle. Similarly 

198 



APPENDIX I 199 

the sine of AOD is less than three times the sine of 
AOB] moreover it is evident that the increase in the sine 
caused by adding the angle COD is less than that caused 
by adding the equal angle BOC. As the angle approaches 
90 degrees, an increase in this angle will produce much 
less increase in the length of the perpendicular DM than 
the same increase of a small angle. The opposite to this 
is true of the cosine of an angle ; as the angle increases 
the cosine diminishes, and at an increasing rate as the 
angle approaches 90 degrees. 

The sine of an angle increases, as we see, from zero to 
unity as the angle increases from zero to 90 degrees, and 
the cosine diminishes from unity to zero as the angle in- 
creases from zero to 90 degrees. The value of the sine, 
cosine, or tangent of any angle can be found from tables 
which have been constructed by calculation. 

All the trigonometrical formulae used in the preceding 
pages are easily deduced from the foregoing. They are 
as follows : 

, sina , cos a 

tan a = ; cot a = — — ; 

cos a sm a 

sin(a ± &) = sin a • cos b±cosa sin&; sin2a=2 sin a • cos a; 
sin (180 —a) = sin a ; sin^ a + cos^ a = l. 

Also in any triangle ABC, = — , BC being the 

sin B AC 

side opposite to the angle A, and AC being opposite to B, 



APPENDIX II 

The following demonstration is based upon that given 
in Heath's " Geometrical Optics '' : ^ 

In the figure, the normal, MN, to the refracting surface 
lies in the plane of the paper represented by ABCD, and 




Fig. 54. 



the plane EFOH is perpendicular to the plane of the 
paper. For the sake of clearness all lines which lie in 



* " Geometrical Optics," Heath, 2d ed., p. 21. 
200 



APPENDIX II 201 

the plane ABCD are drawn as continuous, and all not in 
this plane are drawn as interrupted lines. 

Let RO represent an incident ray which, lies in front 
of the plane ABCD, then OS, the refracted ray, will lie 
behind this plane. If the first medium be air, whose 
index is 1, and if the index of the second medium be n, 
then we shall have the equation 

sin3/0i? = n . sin>SO^. 

Since the incident and refracted rays must lie in the 
plane MRNS, the deviation produced by the refraction is 
in this plane, but we have learned that a deviation in any 
plane may be resolved into two deviations at right angles 
to each other; and therefore a deviation in the plane 
MRNS may be considered as the resultant of tAvo devia- 
tions lying respectively in the plane ABCD and in PQRS 
at right angles to ABCD. To find these resultant de- 
viations, lay off the distances RO and OS such that 
OS = n • RO. From R, which lies in front of the plane 
ABCD, draw i? J/ perpendicular to MN, and i?P perpen- 
dicular to ABCD\ from S, which lies behind the plane 
ABCD, draw in like manner SN and SQ. Then as RMO 
is a right angle, we have MR = RO - sin 3I0R, and from 
the triangle SOX^e have SX= OS • sin >SO:V, or, since 
OS =71' RO, SX= RO • n • sin SON-, but sin 3I0R = 
n- sin S0^\ hence MR = SX. MR is also parallel to 
SN, since both lie in the same plane and are perpen- 
dicular to MX. We see also that PR is parallel to SQ 
and FM to QX; thus the triangles PJfR and XQS are 
equal, and PR = QS and PM= QN. 



202 HANDBOOK OF OPTICS 

Let the angle POE, which the incident ray makes 

with the plane ABCD, be denoted by i^^ and let the 

angle QOS, which the refracted ray makes with this 

plane, be denoted by r^; then PE = BO - sin i^^ and 

QS = OS • sin Ti, or, since PB = QS and OS = 71- BO, 

we have . . 

sm 1^ = 71' sm T^. (1) 

From this we see that there is the same relation be- 
tween ^l and i\ as between the angles of incidence and 
refraction, MOB and SON. 

If the angle POM be denoted by 4, and'QOiV^ by 
T2, we shall have PM= PO • sin ^2 = i20 • cos ^l • sin 2*2, 
and QJSf= QO • sin rg = OS • cos r^ sin 7-2 = n • BO • cos Vi 
sin 7^2- 

From this, since PM= QN, we have 

cos ^l sin i^^n * cos ri sin rg, 

. . cos ?\ . ,ox 

or sm i2 = n ' ^ • sm rg. (2) 

cos ii 

Thus we see that ^2 and ra are connected by the law 
of refraction, the index of the first medium being 1 and 

cos 7' 

that of the second being 71 • -. From equations (1) 

cos ii 

and (2) we can find the deviation of the ray BOS in the 
planes PBQS and 3fPNQ, or ABCD. 

When n is greater than unity, as in glass, it follows 
from (1) that 2\ is greater than rj, and since the greater 

the angle the smaller is the cosine, ~ must be greater 

cos ?*i 

than unity ; hence 71 • — ^ — - is greater than n. In other 

cos 1*1 



APPENDIX II 203 

words, the deviation of the ray in the direction of the 
plane ABCD is the same as would be produced in a 
ray, PO, upon entering a medium of greater index than 
that of the medium which we are considering. As we 
know that the deviation increases with the index, it fol- 
lows that the ray EO undergoes a greater deviation in 
the plane ABCD than would a ray in the line PO, which 
is the projection oi BO on this plane. 

We can now apply these deductions to refraction 
through a prism. The ]3lane ABCD represents a prin- 
cipal plane of the prism; the TSij BO, not in this plane, 
will, at the first face of the prism, undergo deviation in 
the plane ^IBuSfS, and this deviation is equivalent to a 
certain deviation in the principal plane superposed upon 
a deviation in the plane PBQS, which is perpendicular 
to the principal plane. The plane PBQS, being perpen- 
dicular to the principal plane, cuts the two faces of the 
prism in two parallel lines. From equation (1) we have 
seen that the angles BOB and QOS, whose difference 
expresses the deviation in the plane PBQS, are con- 
nected by the law of refraction; and, since this plane 
cuts the faces of the prism in parallel lines, it is clear 
that the deviation in this plane at the first face must be 
neutralized by that at the second face. Thus we see 
that in any position of the ray no deviation is produced 
in the direction of the edge of the prism. ^ 

Let us now investigate the deviation by the prism in 

* We have already assumed this to be true for the cylindrical 
lens, which we may consider as composed of an infinite number 
of prisms whose edges are all parallel to the axis of the lens. 



204 HANDBOOK OF OPTICS 

its principal plane. The edge of the prism would be 
represented by a line perpendicular to ABCD^ such a 
line would be parallel to FB, and consequently B and P 
are equally distant from the edge. At the first face the 
deviation in the principal plane is greater for the ray 
BO than for a ray, PO, equally distant from the edge of 
the prism. We have seen that the deviation in a prin- 
cipal plane which the ray BO undergoes at the first face 
is the same as would be produced in the ray PO upon 

entering a medium whose index is n • — ^. We have 

cos ii 

seen also that in its passage through the prism there is 
no deviation of the ray in the plane PBQS\ hence, if 
we trace the ray backward, we shall have at the second 
face the same angles, POB and QOS, that we have at 
the first face. At the second face also, then, the oblique 
ray is deviated in the principal plane to the same extent 
as the corresponding ray in this plane would be deviated 

by a prism whose index is n • \ Since the total 

cos ii 

deviation produced by a prism increases as the index 
increases, it is proved that the deviation in the principal 
plane is greater for rays not in this plane than for those 
which lie in it. 

When POB (ii) increases, the index of the hypo- 
thetical prism increases at a continually increasing rate. 



INDEX 



Aberration, chromatic, 29. 
spherical, 28, 181. 
of crystalliue lens, 64. 
Accommodation, 95. 

effect of, on retinal images, 118. 
Airy, Sir George, 124. 
Ametropia, 94. 

Angles of incidence and refrac- 
tion, 7. 
Aphakic eye, curvature hyperopia 
of, 120. 
dioptric power of, 100. 
effect on images of correcting 

lens in, 120. 
focal distances of, 100, 120 
with axial elongation, 101. 
Aqueous humor, 62. 

refractive index of, 83. 
Astigmatism, 124, 133. 
effect of, on retinal images, 134. 
lenticular, 161. 

produced by oblique spherical 
refraction, 165. 
the action of a prism, 172. 
Axis of cylindrical lens, 125. 
of optical system, 65. 
refracting surface, 27. 

Bruecke, 189. 

Cardinal planes and points, 37, 65. 

of the eye, 86. 

in combination with a lens, 90. 
Cartesian oval, 29. 
Cataract, vision after extraction 
of, 121. 



Catoptrics, 1. 

Caustic, 28. 

Centrad, 22. 

Centre, optical, 43. 

Circle of least confusion, 132. 

Colors of the spectrum, 17. 

Conjugate focal planes, 36. 

Conjugate focal distances and foci. 

See Focal distances. 
Constants, optical, of the eye, 67. 
Continuous fraction, 69. 
Convergent, 70. 
Cornea, 62. 
curvature of, asymmetrical, 124. 
method of determining, 186. 
normal, 83. 
Corpuscular theory, S. 
Critical angle, 12. 
Crystalline lens, change in curva- 
ture of, in accommodation, 
95. 
curvature of, 83. 
effect of extraction of, in axial 
myopia, 101. 
in curvature myopia, 101. 
in emmetropia, 100. 
on retinal images, 120. 
effect of variation in position of, 

104. 
equivalent refractive index of, 

64. 
normal position of, 83. 
oblique position of, in astigma- 
tism, 124. 
structure of, 62. 
thickness of, 83. 



205 



206 



INDEX 



Cylindrical lens, 125. 
action of, 125. 
axis of, 125. 
combination of, with spherical 

lens, 125. 
lenses, combination of, at right 
angles, 126. 
at oblique angles, 148. 
effect of, on retinal images, 

135, 146. 
twisting property of, 138. 

Dennett, 22. 

Descartes, 8. 

Deviation, increase of, with in- 
crease of angle of incidence, 
14. 

Dioptre, 61. 

Dioptrics, 1. 

Direct method of ophthalmoscopy, 
193. 

Direct pencil, 163. 

Dispersion, 17. 

Donders, 88, 149, 157, 161. 

Double refraction, 187. 

Electro-magnetic theory of waves, 

4. 
Ellipsoidal surfaces, 40. 
Emission theory, 3. 
Emmetropia, 94. 
Errors of refraction, 94. 
Ether, 3. 

Eye, adaptation of, to varying dis- 
tances, 95. 

aphakic, dioptric power of, 100. 
focal distances of, 100, 120. 

artificial, 89. 

as an optical system, 62. 

cardinal points of, 86. 

conjugate foci and focal dis- 
tances of, 98. 

far point of, 97. 

illumination of, 177. 

lengthening of, in myopia, 98. 

nodal points of, 85. 



Eye, normal, dioptric power of, 
100. 
focal distances of, 85. 
optical constants of, 67, 83. 
principal points of, 85. 
reduced, 87. 
schematic, 86, 88. 
shortening of, in hyperopia, 98. 

Fizeau and Foucault, 3. 

Focal distance of cylindrical lens, 

126. 
Focal distances and foci, conju- 
gate, of single spherical re- 
fracting surface, 30. 
of a lens, 45. 

of a system of surfaces, 65. 
in reflexion, 179. 
principal, of single spherical re- 
fracting surface, 32. 
of a lens, 48, 54. 
of a system of surfaces, 78. 
of the eye, 85. 

of the aphakic eye, 100, 121. 
of the eye in accommodation, 
95, 118. 
relation between principal and 
conjugate, 32, 57, 102. 
Focal interval of Sturm, 131. 
Focal length of spectacle lenses, 

54. 
Focal line of cylindrical lens, 126. 
lines of bi-cylindrical lens, 131. 
Focal planes. See Planes. 
Foci. See Focal distances. 
Focus, 27. 
conjugate, 30. 
principal, 32. 
real, 33. 
virtual, 33. 
Fresnel, 3. 

Galileo, 2. 
Gauss, 64, 65. 
Green, John, 164. 



INDEX 



207 



Hay, Gustavus, 149. 
Heath, 150, 161. 
Helmholtz, 62, 64, 79, 184, 188. 
Helmholtz' formula, 39. 
Huyghens, 3. 
Hyperopia, 94. 

effect of correcting lens on im- 
ages in, 118. 

use of convex lenses in, 96. 

Image, formed by astigmatic re- 
fraction, 131. 
by refraction at one spherical 

surface, 36, 37. 
by refraction through a lens, 

52, 58. 
by a system of spherical re- 
fracting surfaces, 79. 
Helmholtz' formula for deter- 
mining size of, 39. 
real, 58. 
reflected, 181. 

relation between length of pencil 
and size of, 170. 
size of, and of object. See 
Object, 
retinal, effect on, of cylindrical 
lens, 135. 
of prism, 169, 173, 174. 
of spherical lens, 109, 117, 118, 
122. 
virtual, 59. 
Incident ray, 7. 
Index of refraction. See Kefrac- 

tive index. 
Indirect method of ophthalmo- 
scopy, 193. 



Jackson, Edward, 22, 149. 
Javal and Schiotz, 186. 



Kepler, 2, 8. 
Knapp, 196. 
Roller, 144. 



Law, Snell's, of refraction, 8. 

of reflexion, 177. 
Lens, 40. 

crystalline. See Crystalline. 
Lenses, action of concave and con- 
vex, 56. 
bi-cylindrical, 127, 148. 
classification of, 40. 
collective, 57. 
combination of, 60. 
cylindrical. See Cylindrical, 
deviating power of, 154. 
dispersive, 57. 
focal length of, 48, 54. 
nodal points of, 41. 
oblique refraction through, 163. 
periscopic, 41, 59. 
piano-curved, 41, 59. 
position of, when used as spec- 
tacles, 97, 99, 104, 118. 
principal planes and points of, 

49. 
refractive power of, 59. 
spherical, effect of, on images, 
109, 117, 118, 122. 
in hyperopia, 96. 
in myopia, 97. 
in presbyopia, 105. 
sphero-cylindrical, 131, 148. 
systems of numbering, 59, 61. 
thickness of, 40. 
thin, 54. 
toric, 125. 
Lens, Stokes, 149. 
Lenticular astigmatism, 161. 
Light, method of transmission of, 3. 
reflected and scattered, 6. 
velocity of, 3. 
in media of different densities, 
8, 17. 
Lippershey, Hans, 2. 
Listing, 65, 87. 

Maddox, 22. 

prism, 184. 
Meniscus, 40. 



208 



INDEX 



Meridian, principal, 124. 
Metre-angle, 22. 
Minimum deviation, 20. 
Moebius, 64. 
Myopia, 94, 95. 
concave lenses in, 97. 

effect of, on images, 119. 
extraction of crystalline, for cure 
of, 101, 120. 

Nagel, 22. 

Near vision, effect of changing 
position of lens in, 104, 122. 
in emmetropia, 94. 
in hyperopia, 95, 105, 119. 
in myopia, 119. 
in presbyopia, 105, 122. 
Newcomb, 3. 
Newton, Sir Isaac, 2. 
Nodal point of single spherical re- 
fracting surface, 37. 
points, of lens, 41, 54. 
of system of refracting sur- 
faces, 81. 
of the eye, 85. 

Object, relation between size of, 
and image, after one spheri- 
cal refraction, 37. 
after refraction by a lens, 52. 
after refraction by a system of 

surfaces, 79. 
in reflexion, 181. 
Oblique pencil, 163. 
Oblique refraction, cylindrical, 167. 

spherical, 163. 
Ophthalmometer, 184. 
Ophthalmometry, optical prin- 
ciples of, 184. 
Ophthalmoscope, 188. 
Ophthalmoscopy, direct, 193. 
indirect, 194. 
optical principles of, 189. 
Optical centre, 43. 
Optic disk, apparent size and 
shape of, 194. 



Optics, definition and subdrvision 
of, 1. 

Paraboloid surfaces, 40. 
Parallel rays, 31. 
Pencil of light, 6. 
direct, 163. 
oblique, 163. 
Planes and points, cardinal. See 
Cardinal, 
conjugate. See Conjugate, 
principal. See Principal. 
Prentice, 22, 149. 

Presbyopes, near vision of, 105,119. 
Principal focal distances and foci. 

See focal distances. 
Principal meridians, 124. 
Principal plane and point, of single 
spherical refracting surface, 
36. 
Principal plane of prism, 14, 169. 
Principal planes and points of a 
lens, 49. 
of a system of surfaces, 72. 
of the eye, 85. 

of the eye in combination with 
a lens, 93. 
Prism, 14. 

Wollaston's, 187. 
Prism-dioptre, 22. 
Prisms, astigmatic effect of, 172. 
deviation produced by, 19, 21. 
effect of, on pencils of light and 

images, 169. 
refracting angle of, 18. 
refraction of pencils by, 169. 
of rays by, 14. 

of rays not in principal plane, 
170. 
combination of, 22. 
numbering of, 21. 
use of, in oiDhthalmology, 17. 

Ray, 6. 
extraordinary and ordinary, 187. 
incident and refracted, 7. 



INDEX 



209 



Reduced eye, 87. 
Reflected images, 181. 

formation of, at surfaces of 
cornea and crystalline lens, 
182. 
Reflexion at a plane surface, 178. 
at a spherical surface, 178. 
law of, 177. 
total internal, 12. 
Refractive index, 7. 
absolute, 13. 
of spectacle glass, 21. 
Refraction by one spherical sur- 
face, 26. 
by cylindrical surfaces, 125. 
by spherical lenses, 41. 
by system of spherical surfaces, 

65. 
by toric surfaces, 125. 
double, 187. 
law of, 8. 
oblique, 163. 
Retinal images, as affected by 
accommodation, 118. 
by cylindrical lenses, 135. 
by prisms, 169. 
by spherical lenses, 109. 
in aphakia, 120. 
in hyperopia, 118. 
in myopia, 119. 
in presbyopia, 119. 
Reversibility of path of light, 12. 
Romer, 2. 



Schematic eye, 86, 88. 
Skiascopy, 191. 
Snell, 2, 8. 
Spectacles, 2, 54. 
Spectrum, colors of, 17. 
Sphero-cylindrical equivalence of 

bi-cj^lindrical lenses, 131, 

148. 
equivalent, practical method of 

finding, 160. 
Stokes, Sir G. G., 148. 
Stokes lens, 149. 
Sturm, focal interval of, 131. 

Toric surface, 124. 

Units used in numbering lenses, 
59, 61. 
prisms, 21. 

Vision, theory of ancients, regard- 
ing, 2. 
Visual angle, 171. 
Vitreous, 64. 
refractive index of, 83. 

Wave theory, 3. 
Weiland 149, 157. 
Wollaston's prism, 187. 

Young, Dr. Thomas, 3, 124. 



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